18.2 Functions that compute some of the sequences in Sloane's tables

Module: sage.combinat.sloane_functions

Functions that compute some of the sequences in Sloane's tables

Type sloane.[tab] to see a list of the sequences that are defined.

sage: a = sloane.A000005; a
 The integer sequence tau(n), which is the number of divisors of n.
 sage: a(1)
 1
 sage: a(6)
 4
 sage: a(100)
 9

Type d._eval?? to see how the function that computes an individual term of the sequence is implemented.

The input must be a positive integer:

sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

You can also change how a sequence prints:

sage: a = sloane.A000005; a
The integer sequence tau(n), which is the number of divisors of n.
sage: a.rename('(..., tau(n), ...)')
sage: a
(..., tau(n), ...)
sage: a.reset_name()
sage: a
The integer sequence tau(n), which is the number of divisors of n.

TESTS:

sage: a = sloane.A000001;
sage: a == loads(dumps(a))
True

Author Log:

William Stein: framework
Jaap Spies: most sequences
Nick Alexander: updated framework

Module-level Functions

perm_mh( m, h)

This functions calculates from Sloane's sequences A079908-A079928

Input:

m: - positive integer
h: - non negative integer

Output: permanent of the m x (m+h) matrix, etc.

sage: from sage.combinat.sloane_functions import perm_mh
sage: perm_mh(3,3)
36
sage: perm_mh(3,4)
76

Author: Jaap Spies (2006)

recur_gen2( a0, a1, a2, a3)

homogenous general second-order linear recurrence generator with fixed coefficients

a(0) = a0, a(1) = a1, a(n) = a2*a(n-1) + a3*a(n-2)

sage: from sage.combinat.sloane_functions import recur_gen2
sage: it = recur_gen2(1,1,1,1)
sage: [it.next() for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

recur_gen2b( a0, a1, a2, a3, b)

inhomogenous second-order linear recurrence generator with fixed coefficients and

, , .

sage: from sage.combinat.sloane_functions import recur_gen2b
sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
sage: [it.next() for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

recur_gen3( a0, a1, a2, a3, a4, a5)

homogenous general third-order linear recurrence generator with fixed coefficients

a(0) = a0, a(1) = a1, a(2) = a2, a(n) = a3*a(n-1) + a4*a(n-2) + a5*a(n-3)

sage: from sage.combinat.sloane_functions import recur_gen3
sage: it = recur_gen3(1,1,1,1,1,1)
sage: [it.next() for i in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

Class: A000001

class A000001

A000001( self)

Number of groups of order .

Note: The database_gap-4.4.9 must be installed for .

run sage -i database_gap-4.4.9 or higher first.

Input:

n: - positive integer

Output: integer

sage: a = sloane.A000001;a
Number of groups of order n.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1) #optional database_gap
1
sage: a(2) #optional database_gap
1
sage: a(9) #optional database_gap
2
sage: a.list(16) #optional database_gap
[1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14]
sage: a(60)     # optional
13

Author: Jaap Spies (2007-02-04)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: sloane.A000001._eval(4)
2
sage: sloane.A000001._eval(51) #optional requires database_gap

_repr_( self)

sage: sloane.A000001._repr_()
'Number of groups of order n.'

Class: A000004

class A000004

A000004( self)

The zero sequence.

Input:

n: - non negative integer

Output:

sage: a = sloane.A000004; a
The zero sequence.
sage: a(1)
0
sage: a(2007)
0
sage: a.list(12)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Author: Jaap Spies (2006-12-10)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: sloane.A000004._eval(5)
0

_repr_( self)

sage: sloane.A000004._repr_()
'The zero sequence.'

Class: A000005

class A000005

A000005( self)

The sequence , which is the number of divisors of .

This sequence is also denoted (also called or $\sigma_0(n)$ ), the number of divisors of n.

Input:

n: - positive integer

Output:

sage: d = sloane.A000005; d
The integer sequence tau(n), which is the number of divisors of n.
sage: d(1)
1
sage: d(6)
4
sage: d(51)
4
sage: d(100)
9
sage: d(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: d.list(10)
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4]

Author Log:

Jaap Spies (2006-12-10)
William Stein (2007-01-08)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: sloane.A000005._eval(5)
2

_repr_( self)

sage: sloane.A000005._repr_()
'The integer sequence tau(n), which is the number of divisors of n.'

Class: A000007

class A000007

A000007( self)

The characteristic function of 0: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000007;a
The characteristic function of 0: a(n) = 0^n.
sage: a(0)
1
sage: a(2)
0
sage: a(12)
0
sage: a.list(12)
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Author: Jaap Spies (2007-01-12)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000007._eval(n) for n in range(10)]
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0]

_repr_( self)

sage: sloane.A000007._repr_()
'The characteristic function of 0: a(n) = 0^n.'

Class: A000009

class A000009

A000009( self)

Number of partitions of into odd parts.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000009;a
Number of partitions of n into odd parts.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
18
sage: a.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

Author: Jaap Spies (2007-01-30)

Functions: cf, $\,$ list

cf( self)

sage: it = sloane.A000009.cf()
sage: [it.next() for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

list( self, n)

sage: sloane.A000009.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000009._eval(i) for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]

_precompute( self, [how_many=50])

sage: initial = len(sloane.A000009._b)
sage: sloane.A000009._precompute(10)
sage: len(sloane.A000009._b) - initial == 10
True

_repr_( self)

sage: sloane.A000009._repr_()
'Number of partitions of n into odd parts.'

Class: A000010

class A000010

A000010( self)

The integer sequence A000010 is Euler's totient function.

Number of positive integers that are relative prime to . Number of totatives of .

Euler totient function $\phi(n)$ : count numbers < and prime to . euler_phi is a standard SAGE function implemented in PARI

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000010; a
Euler's totient function
sage: a(1)
1
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(11)
10
sage: a.list(12)
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2007-01-12)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000010._eval(n) for n in range(1,11)]
       [1, 1, 2, 2, 4, 2, 6, 4, 6, 4]

_repr_( self)

sage: sloane.A000010._repr_()
"Euler's totient function"

Class: A000012

class A000012

A000012( self)

The all 1's sequence.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000012; a
The all 1's sequence.
sage: a(1)
1
sage: a(2007)
1
sage: a.list(12)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Author: Jaap Spies (2007-01-12)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000012._eval(n) for n in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

_repr_( self)

sage: sloane.A000012._repr_()
"The all 1's sequence."

Class: A000015

class A000015

A000015( self)

Smallest prime power $\geq n$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000015; a
Smallest prime power >= n.
sage: a(1)
1
sage: a(8)
8
sage: a(305)
307
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 13]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000015._eval(n) for n in range(1,11)]
       [1, 2, 3, 4, 5, 7, 7, 8, 9, 11]

_repr_( self)

sage: sloane.A000015._repr_()
'Smallest prime power >= n.'

Class: A000016

class A000016

A000016( self)

Sloane's A000016

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000016; a
Sloane's A000016.
sage: a(1)
1
sage: a(0)
1
sage: a(8)
16
sage: a(75)
251859545753048193000
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94]

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000016._eval(n) for n in range(10)]
       [1, 1, 1, 2, 2, 4, 6, 10, 16, 30]

_repr_( self)

sage: sloane.A000016._repr_()
"Sloane's A000016."

Class: A000027

class A000027

A000027( self)

The natural numbers. Also called the whole numbers, the counting numbers or the positive integers.

The following examples are tests of SloaneSequence more than A000027.

sage: s = sloane.A000027; s
The natural numbers.
sage: s(10)
10

Index n is interpreted as _eval(n):

sage: s[10]
10

Slices are interpreted with absolute offsets, so the following returns the terms of the sequence up to but not including the third term:

sage: s[:3]
[1, 2]
sage: s[3:6]
[3, 4, 5]
sage: s.list(5)
[1, 2, 3, 4, 5]

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: sloane.A000027._eval(5)
5

_repr_( self)

sage: sloane.A000027._repr_()
'The natural numbers.'

Class: A000030

class A000030

A000030( self)

Initial digit of .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000030; a
Initial digit of n
sage: a(0)
0
sage: a(1)
1
sage: a(8)
8
sage: a(454)
4
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1]

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000030._eval(n) for n in range(10)]
       [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

_repr_( self)

sage: sloane.A000030._repr_()
'Initial digit of n'

Class: A000032

class A000032

A000032( self)

Lucas numbers (beginning at 2): .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000032; a
Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
sage: a(0)
2
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be an integer >= 0
sage: a.list(12)
[2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199]

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000032._eval(n) for n in range(10)]
       [2, 1, 3, 4, 7, 11, 18, 29, 47, 76]

_repr_( self)

sage: sloane.A000032._repr_()
'Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).'

Class: A000035

class A000035

A000035( self)

A simple periodic sequence.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000035;a
A simple periodic sequence.
sage: a(0.0)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer
sage: a(1)
1
sage: a(2)
0
sage: a(9)
1
sage: a.list(10)
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

Author: Jaap Spies (2007-02-02)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000035._eval(n) for n in range(10)]
       [0, 1, 0, 1, 0, 1, 0, 1, 0, 1]

_repr_( self)

sage: sloane.A000035._repr_()
'A simple periodic sequence.'

Class: A000040

class A000040

A000040( self)

The prime numbers.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000040; a
The prime numbers.
sage: a(1)
2
sage: a(8)
19
sage: a(305)
2011
sage: a.list(12)
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Author: Jaap Spies (2007-01-17)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000040._eval(n) for n in range(1,11)]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]

_repr_( self)

sage: sloane.A000040._repr_()
'The prime numbers.'

Class: A000041

class A000041

A000041( self)

= number of partitions of (the partition numbers).

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000041;a
a(n) = number of partitions of n (the partition numbers).
sage: a(0)
1
sage: a(2)
2
sage: a(8)
22
sage: a(200)
3972999029388
sage: a.list(9)
[1, 1, 2, 3, 5, 7, 11, 15, 22]

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000041._eval(n) for n in range(1,11)]
       [1, 2, 3, 5, 7, 11, 15, 22, 30, 42]

_repr_( self)

sage: sloane.A000041._repr_()
'a(n) = number of partitions of n (the partition numbers).'

Class: A000043

class A000043

A000043( self)

Primes such that is prime. is then called a Mersenne prime.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000043;a
Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne
prime.
sage: a(1)
2
sage: a(2)
3
sage: a(39)
13466917
sage: a(40)
Traceback (most recent call last):
...
IndexError: list index out of range
sage: a.list(12)
[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000043._eval(n) for n in range(1,11)]
       [2, 3, 5, 7, 13, 17, 19, 31, 61, 89]

_repr_( self)

sage: sloane.A000043._repr_()
'Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne
prime.'

Class: A000045

class A000045

A000045( self)

Sequence of Fibonacci numbers, offset 0,4.

REFERENCES: S. Plouffe, Project Gutenberg, The First 1001 Fibonacci Numbers, http://ibiblio.org/pub/docs/books/gutenberg/etext01/fbncc10.txt We have one more. Our first Fibonacci number is 0.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000045; a
Fibonacci numbers with index n >= 0
sage: a(0)
0
sage: a(1)
1
sage: a.list(12)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2007-01-13)

Functions: fib, $\,$ list

fib( self)

Returns a generator over all Fibanacci numbers, starting with 0.

sage: it = sloane.A000045.fib()
       sage: [it.next() for i in range(10)]
       [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

list( self, n)

sage: sloane.A000045.list(10)
       [0, 1, 1, 2, 3, 5, 8, 13, 21, 34]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000045._eval(n) for n in range(1,11)]
       [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

_precompute( self, [how_many=500])

sage: initial = len(sloane.A000045._b)
sage: sloane.A000045._precompute(10)
sage: len(sloane.A000045._b) - initial > 0
True

_repr_( self)

sage: sloane.A000045._repr_()
'Fibonacci numbers with index n >= 0'

Class: A000069

class A000069

A000069( self)

Odious numbers: odd number of 1's in binary expansion.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000069; a
Odious numbers: odd number of 1's in binary expansion.
sage: a(0)
1
sage: a(2)
4
sage: a.list(9)
[1, 2, 4, 7, 8, 11, 13, 14, 16]

Author: Jaap Spies (2007-02-02)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000069._eval(n) for n in range(10)]
       [1, 2, 4, 7, 8, 11, 13, 14, 16, 19]

_repr_( self)

sage: sloane.A000069._repr_()
"Odious numbers: odd number of 1's in binary expansion."

Class: A000073

class A000073

A000073( self)

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, ...

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000073;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(11)
149
sage: a.list(12)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149]

Author: Jaap Spies (2007-01-19)

Functions: list

list( self, n)

sage: sloane.A000073.list(10)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000073._eval(n) for n in range(10)]
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]

_precompute( self, [how_many=20])

sage: initial = len(sloane.A000073._b)
sage: sloane.A000073._precompute(10)
sage: len(sloane.A000073._b) - initial == 10
True

_repr_( self)

sage: sloane.A000073._repr_()
'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'

Class: A000079

class A000079

A000079( self)

Powers of 2: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000079;a
Powers of 2: a(n) = 2^n.
sage: a(0)
1
sage: a(2)
4
sage: a(8)
256
sage: a(100)
1267650600228229401496703205376
sage: a.list(9)
[1, 2, 4, 8, 16, 32, 64, 128, 256]

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000079._eval(n) for n in range(10)]
       [1, 2, 4, 8, 16, 32, 64, 128, 256, 512]

_repr_( self)

sage: sloane.A000079._repr_()
'Powers of 2: a(n) = 2^n.'

Class: A000085

class A000085

A000085( self)

Number of self-inverse permutations on letters, also known as involutions; number of Young tableaux with cells.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000085;a
Number of self-inverse permutations on n letters.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
2
sage: a(12)
140152
sage: a.list(13)
[1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152]

Author: Jaap Spies (2007-02-03)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000085._eval(n) for n in range(10)]
       [1, 1, 2, 4, 10, 26, 76, 232, 764, 2620]

_repr_( self)

sage: sloane.A000085._repr_()
'Number of self-inverse permutations on n letters.'

Class: A000100

class A000100

A000100( self)

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000100;a
Number of compositions of n in which the maximum part size is 3.
sage: a(0)
0
sage: a(1)
0
sage: a(2)
0
sage: a(3)
1
sage: a(11)
360
sage: a.list(12)
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000100._eval(n) for n in range(10)]
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94]

_repr_( self)

sage: sloane.A000100._repr_()
'Number of compositions of n in which the maximum part size is 3.'

Class: A000108

class A000108

A000108( self)

Catalan numbers: $C_n = \frac{{{2n}\choose{n}}}{n+1} = \frac {(2n)!}{n!(n+1)!}$ . Also called Segner numbers.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000108;a
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also
called Segner numbers.
sage: a(0)
1
sage: a.offset
0
sage: a(8)
1430
sage: a(40)
2622127042276492108820
sage: a.list(9)
[1, 1, 2, 5, 14, 42, 132, 429, 1430]

Author: Jaap Spies (2007-01-12)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000108._eval(n) for n in range(10)]
       [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]

_repr_( self)

sage: sloane.A000108._repr_()
'Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!). Also
called Segner numbers.'

Class: A000110

class A000110

A000110( self)

The sequence of Bell numbers.

The Bell number counts the number of ways to put distinguishable things into indistinguishable boxes such that no box is empty.

Let denote the Stirling number of the second kind. Then

$\displaystyle B_n = \sum{k=0}^{n} S(n, k) .$

Input:

n: - integer >= 0

Output:

integer: -

sage: a = sloane.A000110; a
Sequence of Bell numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
475853912767648336587907688413872078263636696868256114666163346375591144978
92442622672724044217756306953557882560751
sage: a.list(10)
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

Author: Nick Alexander

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A000110._repr_()
'Sequence of Bell numbers'

Class: A000120

class A000120

A000120( self)

1's-counting sequence: number of 1's in binary expansion of .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000120;a
1's-counting sequence: number of 1's in binary expansion of n.
sage: a(0)
0
sage: a(2)
1
sage: a(12)
2
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3]

Author: Jaap Spies (2007-01-26)

Functions: f

f( self, n)

sage: [sloane.A000120.f(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000120._eval(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]

_repr_( self)

sage: sloane.A000120._repr_()
"1's-counting sequence: number of 1's in binary expansion of n."

Class: A000124

class A000124

A000124( self)

Central polygonal numbers (the Lazy Caterer's sequence): .

Or, maximal number of pieces formed when slicing a pancake with cuts.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000124;a
Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1.
sage: a(0)
1
sage: a(1)
2
sage: a(2)
4
sage: a(9)
46
sage: a.list(10)
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000124._eval(n) for n in range(10)]
[1, 2, 4, 7, 11, 16, 22, 29, 37, 46]

_repr_( self)

sage: sloane.A000124._repr_()
"Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1."

Class: A000129

class A000129

A000129( self)

Pell numbers: , ; for , .

Denominators of continued fraction convergents to $\sqrt 2$ .

See also A001333

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000129;a
Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
sage: a(0)
0
sage: a(2)
2
sage: a(12)
13860
sage: a.list(12)
[0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A000129._repr_()
'Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).'

Class: A000142

class A000142

A000142( self)

Factorial numbers: $n! = 1 \cdot 2 \cdot 3 \cdots n$

Order of symmetric group , number of permutations of letters.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000142;a
Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number
of permutations of n letters).
sage: a(0)
1
sage: a(8)
40320
sage: a(40)
815915283247897734345611269596115894272000000000
sage: a.list(9)
[1, 1, 2, 6, 24, 120, 720, 5040, 40320]

Author: Jaap Spies (2007-01-12)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000142._eval(n) for n in range(10)]
       [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]

_repr_( self)

sage: sloane.A000142._repr_()
'Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n,
number of permutations of n letters).'

Class: A000153

class A000153

A000153( self)

, with , .

With offset 1, permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000153; a
a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.
sage: a(0)
0
sage: a(1)
1
sage: a(8)
82508
sage: a(20)
10315043624498196944
sage: a.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A000153._repr_()
'a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.'

Class: A000165

class A000165

A000165( self)

Double factorial numbers: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000165;a
Double factorial numbers: (2n)!! = 2^n*n!.
sage: a(0)
1
sage: a.offset
0
sage: a(8)
10321920
sage: a(20)
2551082656125828464640000
sage: a.list(9)
[1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920]

Author: Jaap Spies (2007-01-24)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000165._eval(n) for n in range(10)]
       [1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560]

_repr_( self)

sage: sloane.A000165._repr_()
'Double factorial numbers: (2n)!! = 2^n*n!.'

Class: A000166

class A000166

A000166( self)

Subfactorial or rencontres numbers, or derangements: number of permutations of elements with no fixed points.

With offset 1 also the permanent of a (0,1)-matrix of order with 0's not on a line.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000166;a
Subfactorial or rencontres numbers, or derangements: number of permutations
of $n$ elements with no fixed points.
sage: a(0)
1
sage: a(1)
0
sage: a(2)
1
sage: a.offset
0
sage: a(8)
14833
sage: a(20)
895014631192902121
sage: a.list(9)
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000166._eval(n) for n in range(9)]
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]

_repr_( self)

sage: sloane.A000166._repr_()
'Subfactorial or rencontres numbers, or derangements: number of
permutations of $n$ elements with no fixed points.'

Class: A000169

class A000169

A000169( self)

Number of labeled rooted trees with nodes: $n^{(n-1)}$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000169;a
Number of labeled rooted trees with n nodes: n^(n-1).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(10)
1000000000
sage: a.list(11)
[1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000,
25937424601]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000169._eval(n) for n in range(1,11)]
       [1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000]

_repr_( self)

sage: sloane.A000169._repr_()
'Number of labeled rooted trees with n nodes: n^(n-1).'

Class: A000203

class A000203

A000203( self)

The sequence $\sigma(n)$ , where $\sigma(n)$ is the sum of the divisors of . Also called $\sigma_1(n)$ .

The function sigma(n, k) implements $\sigma_k(n)$ in SAGE.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000203; a
sigma(n) = sum of divisors of n. Also called sigma_1(n).
sage: a(1)
1
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(256)
511
sage: a.list(12)
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000203._eval(n) for n in range(1,11)]
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18]

_repr_( self)

sage: sloane.A000203._repr_()
'sigma(n) = sum of divisors of n. Also called sigma_1(n).'

Class: A000204

class A000204

A000204( self)

Lucas numbers (beginning with 1): with , .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000204; a
Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.
sage: a(1)
1
sage: a(8)
47
sage: a(200)
627376215338105766356982006981782561278127
sage: a(-4)
Traceback (most recent call last):
...
ValueError: input n (=-4) must be a positive integer
sage: a.list(12)
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000204._eval(n) for n in range(1,11)]
[1, 3, 4, 7, 11, 18, 29, 47, 76, 123]

_repr_( self)

sage: sloane.A000204._repr_()
'Lucas numbers (beginning at 1): L(n) = L(n-1) + L(n-2), L(2) = 3.'

Class: A000213

class A000213

A000213( self)

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, ...

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000213;a
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
1
sage: a(11)
355
sage: a.list(12)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355]

Author: Jaap Spies (2007-01-19)

Functions: list

list( self, n)

sage: sloane.A000213.list(10)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000213._eval(n) for n in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]

_precompute( self, [how_many=20])

sage: initial = len(sloane.A000213._b)
sage: sloane.A000213._precompute(10)
sage: len(sloane.A000213._b) - initial == 10
True

_repr_( self)

sage: sloane.A000213._repr_()
'Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3).'

Class: A000217

class A000217

A000217( self)

Triangular numbers: $a(n) = {n+1} \choose 2) = n(n+1)/2$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000217;a
Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.
sage: a(0)
0
sage: a(2)
3
sage: a(8)
36
sage: a(2000)
2001000
sage: a.list(9)
[0, 1, 3, 6, 10, 15, 21, 28, 36]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000217._eval(n) for n in range(10)]
[0, 1, 3, 6, 10, 15, 21, 28, 36, 45]

_repr_( self)

sage: sloane.A000217._repr_()
'Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.'

Class: A000225

class A000225

A000225( self)

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000225;a
2^n - 1.
sage: a(0)
0
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(12)
4095
sage: a.list(12)
[0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000225._eval(n) for n in range(10)]
       [0, 1, 3, 7, 15, 31, 63, 127, 255, 511]

_repr_( self)

sage: sloane.A000225._repr_()
'2^n - 1.'

Class: A000244

class A000244

A000244( self)

Powers of 3: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000244;a
Powers of 3: a(n) = 3^n.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(3)
27
sage: a(11)
177147
sage: a.list(12)
[1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000244._eval(n) for n in range(10)]
       [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]

_repr_( self)

sage: sloane.A000244._repr_()
'Powers of 3: a(n) = 3^n.'

Class: A000255

class A000255

A000255( self)

, with , .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000255;a
a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.
sage: a(0)
1
sage: a(1)
1
sage: a.offset
0
sage: a(8)
148329
sage: a(22)
9923922230666898717143
sage: a.list(9)
[1, 1, 3, 11, 53, 309, 2119, 16687, 148329]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A000255._repr_()
'a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1.'

Class: A000261

class A000261

A000261( self)

, with , .

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000261;a
a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a.offset
1
sage: a(8)
30637
sage: a(22)
1801366114380914335441
sage: a.list(9)
[0, 1, 3, 13, 71, 465, 3539, 30637, 296967]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A000261._repr_()
'a(n) = n*a(n-1) + (n-3)*a(n-2), a(1) = 0, a(2) = 1.'

Class: A000272

class A000272

A000272( self)

Number of labeled rooted trees on nodes: $n^{(n-2)}$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000272;a
Number of labeled rooted trees with n nodes: n^(n-2).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(10)
100000000
sage: a.list(11)
[1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000272._eval(n) for n in range(1,11)]
       [1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000]

_repr_( self)

sage: sloane.A000272._repr_()
'Number of labeled rooted trees with n nodes: n^(n-2).'

Class: A000290

class A000290

A000290( self)

The squares: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000290;a
The squares: a(n) = n^2.
sage: a(0)
0
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(16)
256
sage: a.list(17)
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000290._eval(n) for n in range(10)]
       [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]

_repr_( self)

sage: sloane.A000290._repr_()
'The squares: a(n) = n^2.'

Class: A000292

class A000292

A000292( self)

Tetrahedral (or pyramidal) numbers: ${n+2} \choose 3 = n(n+1)(n+2)/6$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000292;a
Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.
sage: a(0)
0
sage: a(2)
4
sage: a(11)
286
sage: a.list(12)
[0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000292._eval(n) for n in range(10)]
       [0, 1, 4, 10, 20, 35, 56, 84, 120, 165]

_repr_( self)

sage: sloane.A000292._repr_()
'Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6.'

Class: A000302

class A000302

A000302( self)

Powers of 4: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000302;a
Powers of 4: a(n) = 4^n.
sage: a(0)
1
sage: a(1)
4
sage: a(2)
16
sage: a(10)
1048576
sage: a.list(12)
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000302._eval(n) for n in range(10)]
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144]

_repr_( self)

sage: sloane.A000302._repr_()
'Powers of 4: a(n) = 4^n.'

Class: A000312

class A000312

A000312( self)

Number of labeled mappings from points to themselves (endofunctions): .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000312;a
Number of labeled mappings from n points to themselves (endofunctions):
n^n.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(1)
1
sage: a(9)
387420489
sage: a.list(11)
[1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000312._eval(n) for n in range(10)]
       [1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489]

_repr_( self)

sage: sloane.A000312._repr_()
'Number of labeled mappings from n points to themselves (endofunctions):
n^n.'

Class: A000326

class A000326

A000326( self)

Pentagonal numbers: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000326;a
Pentagonal numbers: n(3n-1)/2.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
5
sage: a(10)
145
sage: a.list(12)
[0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000326._eval(n) for n in range(10)]
       [0, 1, 5, 12, 22, 35, 51, 70, 92, 117]

_repr_( self)

sage: sloane.A000326._repr_()
'Pentagonal numbers: n(3n-1)/2.'

Class: A000330

class A000330

A000330( self)

Square pyramidal numbers" $0^2 + 1^2 \cdots n^2 = n(n+1)(2n+1)/6$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000330;a
Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
14
sage: a(11)
506
sage: a.list(12)
[0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000330._eval(n) for n in range(10)]
       [0, 1, 5, 14, 30, 55, 91, 140, 204, 285]

_repr_( self)

sage: sloane.A000330._repr_()
'Square pyramidal numbers: 0^2+1^2+2^2+...+n^2 = n(n+1)(2n+1)/6.'

Class: A000396

class A000396

A000396( self)

Perfect numbers: equal to sum of proper divisors.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000396;a
Perfect numbers: equal to sum of proper divisors.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
28
sage: a(7)
137438691328
sage: a.list(7)
[6, 28, 496, 8128, 33550336, 8589869056, 137438691328]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000396._eval(n) for n in range(1,6)]
       [6, 28, 496, 8128, 33550336]

_repr_( self)

sage: sloane.A000396._repr_()
'Perfect numbers: equal to sum of proper divisors.'

Class: A000578

class A000578

A000578( self)

The cubes: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000578;a
The cubes: n^3
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3)
27
sage: a(11)
1331
sage: a.list(12)
[0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000578._eval(n) for n in range(10)]
       [0, 1, 8, 27, 64, 125, 216, 343, 512, 729]

_repr_( self)

sage: sloane.A000578._repr_()
'The cubes: n^3'

Class: A000583

class A000583

A000583( self)

Fourth powers: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000583;a
Fourth powers: n^4.
sage: a(0.0)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer
sage: a(1)
1
sage: a(2)
16
sage: a(9)
6561
sage: a.list(10)
[0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

Author: Jaap Spies (2007-02-04)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000583._eval(n) for n in range(10)]
       [0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561]

_repr_( self)

sage: sloane.A000583._repr_()
'Fourth powers: n^4.'

Class: A000587

class A000587

A000587( self)

The sequence of Uppuluri-Carpenter numbers.

The Uppuluri-Carpenter number counts the imbalance in the number of ways to put distinguishable things into an even number of indistinguishable boxes versus into an odd number of indistinguishable boxes, such that no box is empty.

Let denote the Stirling number of the second kind. Then

$\displaystyle C_n = \sum{k=0}^{n} (-1)^k S(n, k) .$

Input:

n: - integer >= 0

Output:

integer: -

sage: a = sloane.A000587; a
Sequence of Uppuluri-Carpenter numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
397577026456518507969762382254187048845620355238545130875069912944235105204
434466095862371032124545552161
sage: a.list(10)
[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

Author: Nick Alexander

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A000587._repr_()
'Sequence of Uppuluri-Carpenter numbers'

Class: A000668

class A000668

A000668( self)

Mersenne primes (of form where is a prime).

(See A000043 for the values of .)

Warning: a(39) has 4,053,946 digits!

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000668;a
Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the
values of p.)
sage: a(1)
3
sage: a(2)
7
sage: a(12)
170141183460469231731687303715884105727

Warning: a(39) has 4,053,946 digits!

sage: a(40)
Traceback (most recent call last):
...
IndexError: list index out of range
sage: a.list(8)
[3, 7, 31, 127, 8191, 131071, 524287, 2147483647]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000668._eval(n) for n in range(1,11)]
       [3,
        7,
        31,
        127,
        8191,
        131071,
        524287,
        2147483647,
        2305843009213693951,
        618970019642690137449562111]

_repr_( self)

sage: sloane.A000668._repr_()
'Mersenne primes (of form 2^p - 1 where p is a prime). (See A000043 for the
values of p.)'

Class: A000670

class A000670

A000670( self)

Number of preferential arrangements of labeled elements; or number of weak orders on labeled elements.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000670;a
Number of preferential arrangements of n labeled elements.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
3
sage: a(9)
7087261
sage: a.list(10)
[1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

Author: Jaap Spies (2007-02-03)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000670._eval(n) for n in range(1,10)]
       [1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261]

_repr_( self)

sage: sloane.A000670._repr_()
'Number of preferential arrangements of n labeled elements.'

Class: A000720

class A000720

A000720( self)

, the number of primes $\le n$ . Sometimes called .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000720;a
pi(n), the number of primes <= n. Sometimes called PrimePi(n)
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a(8)
4
sage: a(1000)
168
sage: a.list(12)
[0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000720._eval(n) for n in range(1,11)]
       [0, 1, 2, 2, 3, 3, 4, 4, 4, 4]

_repr_( self)

sage: sloane.A000720._repr_()
'pi(n), the number of primes <= n. Sometimes called PrimePi(n)'

Class: A000796

class A000796

A000796( self)

Decimal expansion of $\pi$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000796;a
Decimal expansion of Pi.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(13)
9
sage: a.list(14)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7]
sage: a(100)
7

Author: Jaap Spies (2007-01-30)

Functions: list, $\,$ pi

list( self, n)

sage: sloane.A000796.list(10)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

pi( self)

Based on a algorithm of Lambert Meertens The ABC-programming language!!!

sage: it = sloane.A000796.pi()
sage: [it.next() for i in range(10)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

       sage: [sloane.A000796._eval(n) for n in range(1,11)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]

_precompute( self, [how_many=1000])

sage: initial = len(sloane.A000796._b)
sage: sloane.A000796._precompute(10)
sage: len(sloane.A000796._b) - initial
10

_repr_( self)

sage: sloane.A000796._repr_()
'Decimal expansion of Pi.'

Class: A000961

class A000961

A000961( self)

Prime powers

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A000961;a
Prime powers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17]

Author: Jaap Spies (2007-01-25)

Functions: list

list( self, n)

       sage: sloane.A000961.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000961._eval(n) for n in range(1,11)]
       [1, 2, 3, 4, 5, 7, 8, 9, 11, 13]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A000961._b)
sage: sloane.A000961._precompute()
sage: len(sloane.A000961._b) - initial > 0
True

_repr_( self)

sage: sloane.A000961._repr_()
'Prime powers.'

Class: A000984

class A000984

A000984( self)

Central binomial coefficients: $2n \choose n = \frac {(2n)!} {(n!)^2}$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A000984;a
Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2
sage: a(0)
1
sage: a(2)
6
sage: a(8)
12870
sage: a.list(9)
[1, 2, 6, 20, 70, 252, 924, 3432, 12870]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000984._eval(n) for n in range(10)]
       [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620]

_repr_( self)

sage: sloane.A000984._repr_()
'Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2'

Class: A001006

class A001006

A001006( self)

Motzkin numbers: number of ways of drawing any number of nonintersecting chords among points on a circle.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001006;a
Motzkin numbers: number of ways of drawing any number of nonintersecting
chords among n points on a circle.
sage: a(0)
1
sage: a(1)
1
sage: a(2)
2
sage: a(12)
15511
sage: a.list(13)
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511]

Author: Jaap Spies (2007-02-02)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001006._eval(n) for n in range(10)]
[1, 1, 2, 4, 9, 21, 51, 127, 323, 835]

_repr_( self)

sage: sloane.A001006._repr_()
'Motzkin numbers: number of ways of drawing any number of nonintersecting
chords among n points on a circle.'

Class: A001045

class A001045

A001045( self)

Jacobsthal sequence: , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001045;a
Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
1
sage: a(11)
683
sage: a.list(12)
[0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001045._repr_()
'Jacobsthal sequence: a(n) = a(n-1) + 2a(n-2).'

Class: A001055

class A001055

A001055( self)

Number of ways of factoring with all factors > 1.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001055;a
Number of ways of factoring n with all factors >1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(9)
2
sage: a.list(16)
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5]

Author: Jaap Spies (2007-02-04)

Functions: nwf

nwf( self, n, m)

sage: sloane.A001055.nwf(4,1)
0
sage: sloane.A001055.nwf(4,2)
1
sage: sloane.A001055.nwf(4,3)
1
sage: sloane.A001055.nwf(4,4)
2

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001055._eval(n) for n in range(1,11)]
[1, 1, 1, 2, 1, 2, 1, 3, 2, 2]

_repr_( self)

sage: sloane.A001055._repr_()
'Number of ways of factoring n with all factors >1.'

Class: A001109

class A001109

A001109( self)

is a triangular number: with , .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001109;a
a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0, a(1)=1
sage: a(0)
0
sage: a(1)
1
sage: a(2)
6
sage: a.offset
0
sage: a(8)
235416
sage: a(60)
1515330104844857898115857393785728383101709300
sage: a.list(9)
[0, 1, 6, 35, 204, 1189, 6930, 40391, 235416]

Author: Jaap Spies (2007-01-24)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001109._repr_()
'a(n)^2 is a triangular number: a(n) = 6*a(n-1) - a(n-2) with a(0)=0,
a(1)=1'

Class: A001110

class A001110

A001110( self)

Numbers that are both triangular and square: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001110; a
Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) + 2.
sage: a(0)
0
sage: a(1)
1
sage: a(8)
55420693056
sage: a(21)
4446390382511295358038307980025
sage: a.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

Author: Jaap Spies (2007-01-19)

Functions: g

g( self, k)

sage: sloane.A001110.g(2)
2
sage: sloane.A001110.g(1)
0

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001110._repr_()
'Numbers that are both triangular and square: a(n) = 34a(n-1) - a(n-2) +
2.'

Class: A001147

class A001147

A001147( self)

Double factorial numbers: $(2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001147;a
Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).
sage: a(0)
1
sage: a.offset
0
sage: a(8)
2027025
sage: a(20)
319830986772877770815625
sage: a.list(9)
[1, 1, 3, 15, 105, 945, 10395, 135135, 2027025]

Author: Jaap Spies (2007-01-24)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001147._eval(n) for n in range(10)]
       [1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425]

_repr_( self)

sage: sloane.A001147._repr_()
'Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1).'

Class: A001157

class A001157

A001157( self)

The sequence $\sigma_2(n)$ , sum of squares of divisors of .

The function sigma(n, k) implements $\sigma_k*$ in SAGE.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001157;a
sigma_2(n): sum of squares of divisors of n
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
5
sage: a(8)
85
sage: a.list(9)
[1, 5, 10, 21, 26, 50, 50, 85, 91]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001157._eval(n) for n in range(1,11)]
[1, 5, 10, 21, 26, 50, 50, 85, 91, 130]

_repr_( self)

sage: sloane.A001157._repr_()
'sigma_2(n): sum of squares of divisors of n'

Class: A001189

class A001189

A001189( self)

Number of degree-n permutations of order exactly 2.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001189;a
Number of degree-n permutations of order exactly 2.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(2)
1
sage: a(12)
140151
sage: a.list(13)
[0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495, 35695, 140151, 568503]

Author: Jaap Spies (2007-02-03)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001189._eval(n) for n in range(1,11)]
       [0, 1, 3, 9, 25, 75, 231, 763, 2619, 9495]

_repr_( self)

sage: sloane.A001189._repr_()
'Number of degree-n permutations of order exactly 2.'

Class: A001221

class A001221

A001221( self)

Number of different prime divisors of

Also called omega(n) or $\omega(n)$ . Maximal number of terms in any factorization of . Number of prime powers that divide .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001221; a
Number of distinct primes dividing n (also called omega(n)).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
1
sage: a(41)
1
sage: a(84792)
3
sage: a.list(12)
[0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2]

Author: - Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001221._eval(n) for n in range(1,10)]
[0, 1, 1, 1, 1, 2, 1, 1, 1]

_repr_( self)

sage: sloane.A001221._repr_()
'Number of distinct primes dividing n (also called omega(n)).'

Class: A001222

class A001222

A001222( self)

Number of prime divisors of (counted with multiplicity).

Also called bigomega(n) or $\Omega(n)$ . Maximal number of terms in any factorization of . Number of prime powers that divide .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001222; a
Number of prime divisors of n (counted with multiplicity).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
3
sage: a(41)
1
sage: a(84792)
5
sage: a.list(12)
[0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3]

Author: - Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001222._eval(n) for n in range(1,10)]
[0, 1, 1, 2, 1, 2, 1, 3, 2]

_repr_( self)

sage: sloane.A001222._repr_()
'Number of prime divisors of n (counted with multiplicity).'

Class: A001227

class A001227

A001227( self)

Number of odd divisors of .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001227; a
Number of odd divisors of n
sage: a.offset
1
sage: a(1)
1
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
3
sage: a(256)
1
sage: a(29)
2
sage: a.list(20)
[1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2]
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Author: - Jaap Spies (2007-01-14)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001227._eval(n) for n in range(1,10)]
[1, 1, 2, 1, 2, 2, 2, 1, 3]

_repr_( self)

sage: sloane.A001227._repr_()
'Number of odd divisors of n'

Class: A001333

class A001333

A001333( self)

Numerators of continued fraction convergents to $\sqrt 2$ .

See also A000129

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001333;a
Numerators of continued fraction convergents to sqrt(2).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
3
sage: a(3)
7
sage: a(11)
8119
sage: a.list(12)
[1, 1, 3, 7, 17, 41, 99, 239, 577, 1393, 3363, 8119]

Author: Jaap Spies (2007-02-01)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001333._repr_()
'Numerators of continued fraction convergents to sqrt(2).'

Class: A001358

class A001358

A001358( self)

Products of two primes.

These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001358;a
Products of two primes.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(8)
22
sage: a(200)
669
sage: a.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]

Author: Jaap Spies (2007-01-25)

Functions: list

list( self, n)

sage: sloane.A001358.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001358._eval(n) for n in range(1,10)]
[4, 6, 9, 10, 14, 15, 21, 22, 25]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A001358._b)
sage: sloane.A001358._precompute()
sage: len(sloane.A001358._b) - initial > 0
True

_repr_( self)

sage: sloane.A001358._repr_()
'Products of two primes.'

Class: A001405

class A001405

A001405( self)

Central binomial coefficients: $n \choose \lfloor \frac {n}{ 2} \rfloor$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001405;a
Central binomial coefficients: C(n,floor(n/2)).
sage: a(0)
1
sage: a(2)
2
sage: a(12)
924
sage: a.list(12)
[1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001405._eval(n) for n in range(10)]
       [1, 1, 2, 3, 6, 10, 20, 35, 70, 126]

_repr_( self)

sage: sloane.A001405._repr_()
'Central binomial coefficients: C(n,floor(n/2)).'

Class: A001477

class A001477

A001477( self)

The nonnegative integers.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001477;a
The nonnegative integers.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(3382789)
3382789
sage: a(11)
11
sage: a.list(12)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001477._eval(n) for n in range(10)]
       [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

_repr_( self)

sage: sloane.A001477._repr_()
'The nonnegative integers.'

Class: A001694

class A001694

A001694( self)

This function returns the -th Powerful Number:

A positive integer is powerful if for every prime dividing , also divides .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001694; a
Powerful Numbers (also called squarefull, square-full or 2-full numbers).
sage: a.offset
1
sage: a(1)
1
sage: a(4)
9
sage: a(100)
3136
sage: a(156)
7225
sage: a.list(19)
[1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128,
144]
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Author: Jaap Spies (2007-01-14)

Functions: is_powerful, $\,$ list

is_powerful( self, n)

This function returns True if and only if is a Powerful Number:

A positive integer is powerful if for every prime dividing , also divides . See Sloane's OEIS A001694.

Input:

n: - integer

Output:

True: - if is a Powerful number, else False

sage: a = sloane.A001694
sage: a.is_powerful(2500)
True
sage: a.is_powerful(20)
False

Author: - Jaap Spies (2006-12-07)

list( self, n)

sage: sloane.A001694.list(9)
[1, 4, 8, 9, 16, 25, 27, 32, 36]

Special Functions: __init__, $\,$ _eval, $\,$ _powerful_numbers_in_range, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001694._eval(n) for n in range(1,10)]
[1, 4, 8, 9, 16, 25, 27, 32, 36]

_powerful_numbers_in_range( self, n, m)

sage: sloane.A001694._powerful_numbers_in_range(0,50)
[4, 8, 9, 16, 25, 27, 32, 36, 49]

_precompute( self, [how_many=10000])

sage: initial = len(sloane.A001694._b)
sage: sloane.A001694._precompute()
sage: len(sloane.A001694._b) - initial > 0
True

_repr_( self)

sage: sloane.A001694._repr_()
'Powerful Numbers (also called squarefull, square-full or 2-full numbers).'

Class: A001836

class A001836

A001836( self)

Numbers such that $\phi(2n-1) < \phi(2n)$ , where $\phi$ is Euler's totient function.

Eulers totient function is also known as euler_phi, euler_phi is a standard SAGE function.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A001836; a
Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient
function A000010.
sage: a.offset
1
sage: a(1)
53
sage: a(8)
683
sage: a(300)
17798
sage: a.list(12)
[53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Compare: Searching Sloane's online database... Numbers n such that phi(2n-1) < phi(2n), where phi is Eler's totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]

Author: Jaap Spies (2007-01-17)

Functions: list

list( self, n)

sage: sloane.A001836.list(9)
[53, 83, 158, 263, 293, 368, 578, 683, 743]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001836._eval(n) for n in range(1,10)]
[53, 83, 158, 263, 293, 368, 578, 683, 743]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A001836._b)
sage: sloane.A001836._precompute()
sage: len(sloane.A001836._b) - initial > 0
True

_repr_( self)

sage: sloane.A001836._repr_()
"Numbers n such that phi(2n-1) < phi(2n), where phi is Euler's totient
function A000010."

Class: A001906

class A001906

A001906( self)

bisection of Fibonacci sequence: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001906; a
F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
sage: a(0) 
0       
sage: a(1)
1
sage: a(8)
987
sage: a(22)
701408733
sage: a.list(12)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001906._repr_()
'F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).'

Class: A001909

class A001909

A001909( self)

, with , .

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer >= 2

Output:

integer: - function value

sage: a = sloane.A001909;a
a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.
sage: a(1)
Traceback (most recent call last):
...
ValueError: input n (=1) must be an integer >= 2
sage: a.offset
2
sage: a(2)
0
sage: a(8)
8544
sage: a(22)
470033715095287415734
sage: a.list(9)
[0, 1, 4, 21, 134, 1001, 8544, 81901, 870274]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001909._repr_()
'a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.'

Class: A001910

class A001910

A001910( self)

, with , .

Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer >= 3

Output:

integer: - function value

sage: a = sloane.A001910;a
a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be an integer >= 3
sage: a(3)
0
sage: a.offset
3
sage: a(8)
1909
sage: a(22)
98125321641110663023
sage: a.list(9)
[0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A001910._repr_()
'a(n) = n*a(n-1) + (n-5)*a(n-2), a(3) = 0, a(4) = 1.'

Class: A001969

class A001969

A001969( self)

Evil numbers: even number of 1's in binary expansion.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A001969;a
Evil numbers: even number of 1's in binary expansion.
sage: a(0)
0
sage: a(1)
3
sage: a(2)
5
sage: a(12)
24
sage: a.list(13)
[0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24]

Author: Jaap Spies (2007-02-02)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A001969._eval(n) for n in range(10)]
       [0, 3, 5, 6, 9, 10, 12, 15, 17, 18]

_repr_( self)

sage: sloane.A001969._repr_()
"Evil numbers: even number of 1's in binary expansion."

Class: A002110

class A002110

A002110( self)

Primorial numbers (first definition): product of first primes. Sometimes written $p\char93$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A002110;a
Primorial numbers (first definition): product of first n primes. Sometimes
written p#.
sage: a(0)
1
sage: a(2)
6
sage: a(8)
9699690
sage: a(17)
1922760350154212639070
sage: a.list(9)
[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A002110._eval(n) for n in range(10)]
       [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870]

_repr_( self)

sage: sloane.A002110._repr_()
'Primorial numbers (first definition): product of first n primes. Sometimes
written p#.'

Class: A002113

class A002113

A002113( self)

Palindromes in base 10.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A002113;a
Palindromes in base 10.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
2
sage: a(12)
33
sage: a.list(13)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33]

Author: Jaap Spies (2007-02-02)

Functions: list

list( self, n)

sage: sloane.A002113.list(15)
       [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A002113._eval(n) for n in range(10)]
       [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A002113._b)
sage: sloane.A002113._precompute()
sage: len(sloane.A002113._b) - initial > 0
True

_repr_( self)

sage: sloane.A002113._repr_()
'Palindromes in base 10.'

Class: A002275

class A002275

A002275( self)

Repunits: $\frac {(10^n - 1)}{9}$ . Often denoted by .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A002275;a
Repunits: (10^n - 1)/9. Often denoted by R_n.
sage: a(0)
0
sage: a(2)
11
sage: a(8)
11111111
sage: a(20)
11111111111111111111
sage: a.list(9)
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A002275._eval(n) for n in range(10)]
[0, 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111]

_repr_( self)

sage: sloane.A002275._repr_()
'Repunits: (10^n - 1)/9. Often denoted by R_n.'

Class: A002378

class A002378

A002378( self)

Oblong (or pronic, or heteromecic) numbers: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A002378;a
Oblong (or pronic, or heteromecic) numbers: n(n+1).
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
0
sage: a(1)
2
sage: a(11)
132
sage: a.list(12)
[0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A002378._eval(n) for n in range(10)]
       [0, 2, 6, 12, 20, 30, 42, 56, 72, 90]

_repr_( self)

sage: sloane.A002378._repr_()
'Oblong (or pronic, or heteromecic) numbers: n(n+1).'

Class: A002620

class A002620

A002620( self)

Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, $\lfloor n^2/4 \rfloor$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A002620;a
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(10)
25
sage: a.list(12)
[0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A002620._eval(n) for n in range(10)]
       [0, 0, 1, 2, 4, 6, 9, 12, 16, 20]

_repr_( self)

sage: sloane.A002620._repr_()
'Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).'

Class: A002808

class A002808

A002808( self)

The composite numbers: numbers of the form for and .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A002808;a
The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
6
sage: a(11)
20
sage: a.list(12)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21]

Author: Jaap Spies (2007-01-26)

Functions: list

list( self, n)

sage: sloane.A002808.list(10)
       [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A002808._eval(n) for n in range(1,11)]
       [4, 6, 8, 9, 10, 12, 14, 15, 16, 18]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A002808._b)
sage: sloane.A002808._precompute()
sage: len(sloane.A002808._b) - initial > 0
True

_repr_( self)

sage: sloane.A002808._repr_()
'The composite numbers: numbers n of the form x*y for x > 1 and y > 1.'

Class: A003418

class A003418

A003418( self)

Least common multiple (or lcm) of $\{1, 2, \cdots, n\}$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A003418;a
Least common multiple (or lcm) of {1, 2, ..., n}.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
360360
sage: a.list(14)
[1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360]
sage: a(20.0)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2007-01-31)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A003418._eval(n) for n in range(1,11)]
[1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520]

_repr_( self)

sage: sloane.A003418._repr_()
'Least common multiple (or lcm) of {1, 2, ..., n}.'

Class: A004086

class A004086

A004086( self)

Read n backwards (referred to as in many sequences).

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A004086;a
Read n backwards (referred to as R(n) in many sequences).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
2
sage: a(3333)
3333
sage: a(12345)
54321
sage: a.list(13)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21]

Author: Jaap Spies (2007-02-02)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A004086._eval(n) for n in range(10)]
       [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

_repr_( self)

sage: sloane.A004086._repr_()
'Read n backwards (referred to as R(n) in many sequences).'

Class: A004526

class A004526

A004526( self)

The nonnegative integers repeated`

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A004526;a
The nonnegative integers repeated.
sage: a(0)
0
sage: a(1)
0
sage: a(2)
1
sage: a(10)
5
sage: a.list(12)
[0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A004526._eval(n) for n in range(10)]
       [0, 0, 1, 1, 2, 2, 3, 3, 4, 4]

_repr_( self)

sage: sloane.A004526._repr_()
'The nonnegative integers repeated.'

Class: A005100

class A005100

A005100( self)

Deficient numbers: $\sigma(n) < 2n$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A005100;a
Deficient numbers: sigma(n) < 2n
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(12)
14
sage: a.list(12)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14]

Author: Jaap Spies (2007-01-26)

Functions: list

list( self, n)

sage: sloane.A005100.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A005100._eval(n) for n in range(1,10)]
       [1, 2, 3, 4, 5, 7, 8, 9, 10]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A005100._b)
sage: sloane.A005100._precompute()
sage: len(sloane.A005100._b) - initial > 0
True

_repr_( self)

sage: sloane.A005100._repr_()
'Deficient numbers: sigma(n) < 2n'

Class: A005101

class A005101

A005101( self)

Abundant numbers (sum of divisors of exceeds ).

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A005101;a
Abundant numbers (sum of divisors of n exceeds 2n).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
12
sage: a(2)
18
sage: a(12)
60
sage: a.list(12)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60]

Author: Jaap Spies (2007-01-26)

Functions: list

list( self, n)

sage: sloane.A005101.list(10)
       [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A005101._eval(n) for n in range(1,11)]
       [12, 18, 20, 24, 30, 36, 40, 42, 48, 54]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A005101._b)
sage: sloane.A005101._precompute()
sage: len(sloane.A005101._b) - initial > 0
True

_repr_( self)

sage: sloane.A005101._repr_()
'Abundant numbers (sum of divisors of n exceeds 2n).'

Class: A005117

class A005117

A005117( self)

Square-free numbers

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A005117;a
Square-free numbers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
2
sage: a(12)
17
sage: a.list(12)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17]

Author: Jaap Spies (2007-01-25)

Functions: list

list( self, n)

sage: sloane.A005117.list(10)
       [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A005117._eval(n) for n in range(1,11)]
       [1, 2, 3, 5, 6, 7, 10, 11, 13, 14]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A005117._b)
sage: sloane.A005117._precompute()
sage: len(sloane.A005117._b) - initial > 0
True

_repr_( self)

sage: sloane.A005117._repr_()
'Square-free numbers.'

Class: A005408

class A005408

A005408( self)

The odd numbers a(n) = 2n + 1.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A005408;a
The odd numbers a(n) = 2n + 1.
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: a(0)
1
sage: a(4)
9
sage: a(11)
23
sage: a.list(12)
[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]

Author: Jaap Spies (2007-01-26)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A005408._eval(n) for n in range(10)]
       [1, 3, 5, 7, 9, 11, 13, 15, 17, 19]

_repr_( self)

sage: sloane.A005408._repr_()
'The odd numbers a(n) = 2n + 1.'

Class: A005843

class A005843

A005843( self)

The even numbers: .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A005843;a
The even numbers: a(n) = 2n.
sage: a(0.0)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer
sage: a(1)
2
sage: a(2)
4
sage: a(9)
18
sage: a.list(10)
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

Author: Jaap Spies (2007-02-03)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A005843._eval(n) for n in range(10)]
       [0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

_repr_( self)

sage: sloane.A005843._repr_()
'The even numbers: a(n) = 2n.'

Class: A006318

class A006318

A006318( self)

Large Schroeder numbers.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A006318;a
Large Schroeder numbers.
sage: a(0)
1
sage: a(1)
2
sage: a(2)
6
sage: a(9)
206098
sage: a.list(10)
[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

Author: Jaap Spies (2007-02-03)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A006318._eval(n) for n in range(10)]
       [1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098]

_repr_( self)

sage: sloane.A006318._repr_()
'Large Schroeder numbers.'

Class: A006530

class A006530

A006530( self)

Largest prime dividing (with ).

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A006530;a
Largest prime dividing n (with a(1)=1).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(8)
2
sage: a(11)
11
sage: a.list(15)
[1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5]

Author: Jaap Spies (2007-01-25)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A006530._eval(n) for n in range(1,11)]
       [1, 2, 3, 2, 5, 3, 7, 2, 3, 5]

_repr_( self)

sage: sloane.A006530._repr_()
'Largest prime dividing n (with a(1)=1).'

Class: A006882

class A006882

A006882( self)

Double factorials : $a(n)=n \cdot a(n-2)$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A006882;a
Double factorials n!!: a(n)=n*a(n-2).
sage: a(0)
1
sage: a(2)
2
sage: a(8)
384
sage: a(20)
3715891200
sage: a.list(9)
[1, 1, 2, 3, 8, 15, 48, 105, 384]

Author: Jaap Spies (2007-01-24)

Functions: df, $\,$ list

df( self)

Double factorials n!!: a(n)=n*a(n-2).

sage: it = sloane.A006882.df()
       sage: [it.next() for i in range(10)]
       [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

list( self, n)

sage: sloane.A006882.list(10)
       [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A006882._eval(n) for n in range(10)]
       [1, 1, 2, 3, 8, 15, 48, 105, 384, 945]

_precompute( self, [how_many=10])

sage: initial = len(sloane.A006882._b)
sage: sloane.A006882._precompute(10)
sage: len(sloane.A006882._b) - initial == 10
True

_repr_( self)

sage: sloane.A006882._repr_()
'Double factorials n!!: a(n)=n*a(n-2).'

Class: A007318

class A007318

A007318( self)

Pascal's triangle read by rows: $C(n,k) = {n \choose k} = \frac {n!} {(k!(n-k)!)}$ , $0 \le k \le n$ .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A007318
sage: a(0)
1
sage: a(1)
1
sage: a(13)
4
sage: a.list(15)
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1]
sage: a(100)
715

Author: Jaap Spies (2007-01-31)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A007318._eval(n) for n in range(10)]
[1, 1, 1, 1, 2, 1, 1, 3, 3, 1]

_repr_( self)

sage: sloane.A007318._repr_()
"Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!),
0<=k<=n."

Class: A008275

class A008275

A008275( self)

Triangle of Stirling numbers of first kind, , $n \ge 1$ , $1 \le k \le n$ .

The unsigned numbers are also called Stirling cycle numbers:

$\vert s(n,k)\vert$ = number of permutations of objects with exactly cycles.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A008275;a
Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
-1
sage: a(3)
1
sage: a(11)
24
sage: a.list(12)
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1, 24, -50]

Author: Jaap Spies (2007-02-02)

Functions: s

s( self, n, k)

sage: sloane.A008275.s(4,2)
11
       sage: sloane.A008275.s(5,2)
       -50
       sage: sloane.A008275.s(5,3)
       35

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A008275._eval(n) for n in range(1, 11)]
[1, -1, 1, 2, -3, 1, -6, 11, -6, 1]

_repr_( self)

sage: sloane.A008275._repr_()
'Triangle of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n.'

Class: A008277

class A008277

A008277( self)

Triangle of Stirling numbers of 2nd kind, , $n \ge 1$ , $1 \le k \le n$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A008277;a
Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
1
sage: a(3)
1
sage: a(4.0)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer
sage: a.list(15)
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1]

Author: Jaap Spies (2007-01-31)

Functions: s2

s2( self, n, k)

Returns the Stirling number S2(n,k) of the 2nd kind.

sage: sloane.A008277.s2(4,2)
7

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A008277._eval(n) for n in range(1,11)]
[1, 1, 1, 1, 3, 1, 1, 7, 6, 1]

_repr_( self)

sage: sloane.A008277._repr_()
'Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.'

Class: A008683

class A008683

A008683( self)

Moebius (or Möbius) function $\mu(n)$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A008683;a
Moebius function mu(n).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
-1
sage: a(12)
0
sage: a.list(12)
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A008683._eval(n) for n in range(1,11)]
[1, -1, -1, 0, -1, 1, -1, 0, 0, 1]

_repr_( self)

sage: sloane.A008683._repr_()
'Moebius function mu(n).'

Class: A010060

class A010060

A010060( self)

Thue-Morse sequence.

Let denote the first terms; then , and for $k \ge 0$ , $A_{k+1} = A_k B_k$ , where is obtained from by interchanging 0's and 1's.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A010060;a
Thue-Morse sequence.
sage: a(0)
0
sage: a(1)
1
sage: a(2)
1
sage: a(12)
0
sage: a.list(13)
[0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0]

Author: Jaap Spies (2007-02-02)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A010060._eval(n) for n in range(10)]
       [0, 1, 1, 0, 1, 0, 0, 1, 1, 0]

_repr_( self)

sage: sloane.A010060._repr_()
'Thue-Morse sequence.'

Class: A015521

class A015521

A015521( self)

Linear 2nd order recurrence, , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A015521; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
13107
sage: a(41)
967140655691703339764941
sage: a.list(12)
[0, 1, 3, 13, 51, 205, 819, 3277, 13107, 52429, 209715, 838861]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A015521._repr_()
'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 4 a(n-2).'

Class: A015523

class A015523

A015523( self)

Linear 2nd order recurrence, , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A015523; a
Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(8)
17727
sage: a(41)
6173719566474529739091481
sage: a.list(12)
[0, 1, 3, 14, 57, 241, 1008, 4229, 17727, 74326, 311613, 1306469]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A015523._repr_()
'Linear 2nd order recurrence, a(n) = 3 a(n-1) + 5 a(n-2).'

Class: A015530

class A015530

A015530( self)

Linear 2nd order recurrence, , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A015530;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
4
sage: a.offset
0
sage: a(8)
41008
sage: a.list(9)
[0, 1, 4, 19, 88, 409, 1900, 8827, 41008]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A015530._repr_()
'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 3 a(n-2).'

Class: A015531

class A015531

A015531( self)

Linear 2nd order recurrence, , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A015531;a
Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
4
sage: a.offset
0
sage: a(8)
65104
sage: a(60)
144560289664733924534327040115992228190104
sage: a.list(9)
[0, 1, 4, 21, 104, 521, 2604, 13021, 65104]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A015531._repr_()
'Linear 2nd order recurrence, a(n) = 4 a(n-1) + 5 a(n-2).'

Class: A015551

class A015551

A015551( self)

Linear 2nd order recurrence, , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A015551;a
Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).
sage: a(0)
0
sage: a(1)
1
sage: a(2)
6
sage: a.offset
0
sage: a(8)
570216
sage: a(60)
7110606606530059736761484557155863822531970573036
sage: a.list(9)
[0, 1, 6, 41, 276, 1861, 12546, 84581, 570216]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A015551._repr_()
'Linear 2nd order recurrence, a(n) = 6 a(n-1) + 5 a(n-2).'

Class: A018252

class A018252

A018252( self)

The nonprime numbers, starting with 1.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A018252;a
The nonprime numbers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
4
sage: a(9)
15
sage: a.list(10)
[1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

Author: Jaap Spies (2007-02-04)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A018252._eval(n) for n in range(1,11)]
       [1, 4, 6, 8, 9, 10, 12, 14, 15, 16]

_repr_( self)

sage: sloane.A018252._repr_()
'The nonprime numbers.'

Class: A020639

class A020639

A020639( self)

Least prime dividing with .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A020639;a
Least prime dividing n (a(1)=1).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(13)
13
sage: a.list(14)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2]

Author: Jaap Spies (2007-01-25)

Functions: list

list( self, n)

sage: sloane.A020639.list(10)
       [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A020639._eval(n) for n in range(1,11)]
       [1, 2, 3, 2, 5, 2, 7, 2, 3, 2]

_precompute( self, [how_many=50])

sage: initial = len(sloane.A020639._b)
sage: sloane.A020639._precompute(10)
sage: len(sloane.A020639._b) - initial == 10
True

_repr_( self)

sage: sloane.A020639._repr_()
'Least prime dividing n (a(1)=1).'

Class: A046660

class A046660

Excess of

= number of prime divisors (with multiplicity) - number of prime divisors (without multiplicity).

$\Omega(n) - \omega(n)$ .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A046660; a
Excess of n = Bigomega (with multiplicity) - omega (without multiplicity).
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
0
sage: a(8)
2
sage: a(41)
0
sage: a(84792)
2
sage: a.list(12)
[0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1]

Author: - Jaap Spies (2007-01-19)

Special Functions: _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A046660._eval(n) for n in range(1,10)]
[0, 0, 0, 1, 0, 0, 0, 2, 1]

_repr_( self)

sage: sloane.A046660._repr_()
'Excess of n = Bigomega (with multiplicity) - omega (without
multiplicity).'

Class: A049310

class A049310

A049310( self)

Triangle of coefficients of Chebyshev's : $U(n, \frac x 2)$ polynomials (exponents in increasing order).

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A049310;a
Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials
(exponents in increasing order).
sage: a(0)
1
sage: a(1)
0
sage: a(13)
0
sage: a.list(15)
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1]
sage: a(200)
0
sage: a.keyword
['sign', 'tabl', 'nice', 'easy', 'core', 'triangle']

Author: Jaap Spies (2007-01-31)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A049310._eval(n) for n in range(10)]
[1, 0, 1, -1, 0, 1, 0, -2, 0, 1]

_repr_( self)

sage: sloane.A049310._repr_()
"Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials
(exponents in increasing order)."

Class: A051959

class A051959

A051959( self)

Linear second order recurrence. A051959.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A051959; a
Linear second order recurrence. A051959.
sage: a(0)
1
sage: a(1)
10
sage: a(8)
9969
sage: a(41)
42834431872413650
sage: a.list(12)
[1, 10, 36, 104, 273, 686, 1688, 4112, 9969, 24114, 58268, 140728]

Author: Jaap Spies (2007-01-19)

Functions: g

g( self, k)

sage: sloane.A051959.g(2)
15
sage: sloane.A051959.g(1)
0

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A051959._repr_()
'Linear second order recurrence. A051959.'

Class: A055790

class A055790

A055790( self)

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A055790;a
a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].
sage: a(0)
0
sage: a(1)
2
sage: a(2)
4
sage: a.offset
0
sage: a(8)
165016
sage: a(22)
10356214297533070441564
sage: a.list(9)
[0, 2, 4, 14, 64, 362, 2428, 18806, 165016]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A055790._repr_()
'a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2].'

Class: A061084

class A061084

A061084( self)

Fibonacci-type sequence based on subtraction: , and .

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A061084; a
Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n) =
a(n-2)-a(n-1).
sage: a(0)
1
sage: a(1)
2
sage: a(8)
-29
sage: a(22)
-24476
sage: a.list(12)
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47, -76, 123]
sage: a.keyword
['sign', 'easy', 'nice']

Author: Jaap Spies (2007-01-18)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A061084._eval(n) for n in range(10)]
[1, 2, -1, 3, -4, 7, -11, 18, -29, 47]

_repr_( self)

sage: sloane.A061084._repr_()
'Fibonacci-type sequence based on subtraction: a(0) = 1, a(1) = 2 and a(n)
= a(n-2)-a(n-1).'

Class: A064553

class A064553

A064553( self)

, for and $a(u \cdot v) = a(u) \cdot a(v)$ for .

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A064553;a
a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
1
sage: a(2)
2
sage: a(9)
9
sage: a.list(16)
[1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16]

Author: Jaap Spies (2007-02-04)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A064553._eval(n) for n in range(1,11)]
       [1, 2, 3, 4, 4, 6, 5, 8, 9, 8]

_repr_( self)

sage: sloane.A064553._repr_()
'a(1) = 1, a(prime(i)) = i+1 for i > 0 and a(u*v) = a(u)*a(v) for u,v > 0'

Class: A079922

class A079922

function returns solutions to the Dancing School problem with

girls and

boys.

The value is , the permanent of the (0,1)-matrix of size $n \times n+3$ with if and only if $i \le j \le i+n$ .

REFERENCES: Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A079922; a
Solutions to the Dancing School problem with n girls and n+3 boys
sage: a.offset
1
sage: a(1)
4
sage: a(8)
2227
sage: a.list(8)
[4, 13, 36, 90, 212, 478, 1044, 2227]

Compare: Searching Sloane's online database... Solution to the Dancing School Problem with n girls and n+3 boys: f(n,3). [4, 13, 36, 90, 212, 478, 1044, 2227]

sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Author: - Jaap Spies (2007-01-14)

Special Functions: _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A079922._eval(n) for n in range(1,5)]
[4, 13, 36, 90]

_repr_( self)

sage: sloane.A079922._repr_()
'Solutions to the Dancing School problem with n girls and n+3 boys'

Class: A079923

class A079923

function returns solutions to the Dancing School problem with

girls and

boys.

The value is , the permanent of the (0,1)-matrix of size $n \times n+3$ with if and only if $i \le j \le i+n$ .

REFERENCES: Jaap Spies, Nieuw Archief voor Wiskunde, 5/7 nr 4, December 2006

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A079923; a
Solutions to the Dancing School problem with n girls and n+4 boys
sage: a.offset
1
sage: a(1)
5
sage: a(8)
15458
sage: a.list(8)
[5, 21, 76, 246, 738, 2108, 5794, 15458]

Compare: Searching Sloane's online database... Solution to the Dancing School Problem with n girls and n+4 boys: f(n,4). [5, 21, 76, 246, 738, 2108, 5794, 15458]

sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer

Author: - Jaap Spies (2007-01-17)

Special Functions: _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A079923._eval(n) for n in range(1,11)]
[5, 21, 76, 246, 738, 2108, 5794, 15458, 40296, 103129]

_repr_( self)

sage: sloane.A079923._repr_()
'Solutions to the Dancing School problem with n girls and n+4 boys'

Class: A082411

class A082411

A082411( self)

Second-order linear recurrence sequence with .

, . This is the second-order linear recurrence sequence with and co- prime, that R. L. Graham in 1964 stated did not contain any primes.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A082411;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
76343678551
sage: a(2)
483732902969
sage: a(3)
560076581520
sage: a(20)
2219759332689173
sage: a.list(4)
[407389224418, 76343678551, 483732902969, 560076581520]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A082411._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'

Class: A083103

class A083103

A083103( self)

Second-order linear recurrence sequence with .

, . This is the second-order linear recurrence sequence with and co- prime, that R. L. Graham in 1964 stated did not contain any primes. It has not been verified. Graham made a mistake in the calculation that was corrected by D. E. Knuth in 1990.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A083103;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
1059683225053915111058165141686995
sage: a(2)
2846455926982717743326880272142788
sage: a(3)
3906139152036632854385045413829783
sage: a.offset
0
sage: a(8)
45481392851206651551714764671352204
sage: a(20)
14639253684254059531823985143948191708
sage: a.list(4)
[1786772701928802632268715130455793, 1059683225053915111058165141686995,
2846455926982717743326880272142788, 3906139152036632854385045413829783]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A083103._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'

Class: A083104

class A083104

A083104( self)

Second-order linear recurrence sequence with .

, . This is the second-order linear recurrence sequence with and co-prime. It was found by Ronald Graham in 1990.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A083104;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(3)
3351693458175078679851381267428333
sage: a.offset
0
sage: a(8)
36021870400834012982120004949074404
sage: a(20)
11601914177621826012468849361236300628

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A083104._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'

Class: A083105

class A083105

A083105( self)

Second-order linear recurrence sequence with .

, . This is the second-order linear recurrence sequence with and co-prime. It was found by Donald Knuth in 1990.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A083105;a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(1)
49463435743205655
sage: a(2)
112101715747445512
sage: a(3)
161565151490651167
sage: a.offset
0
sage: a(8)
1853029790662436896
sage: a(20)
596510791500513098192
sage: a.list(4)
[62638280004239857, 49463435743205655, 112101715747445512,
161565151490651167]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A083105._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'

Class: A083216

class A083216

A083216( self)

Second-order linear recurrence sequence with .

, . This is a second-order linear recurrence sequence with and co-prime that does not contain any primes. It was found by Herbert Wilf in 1990.

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A083216; a
Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).
sage: a(0)
20615674205555510
sage: a(1)
3794765361567513
sage: a(8)
347693837265139403
sage: a(41)
2738025383211084205003383
sage: a.list(4)
[20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536]

Author: Jaap Spies (2007-01-19)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A083216._repr_()
'Second-order linear recurrence sequence with a(n) = a(n-1) + a(n-2).'

Class: A090010

class A090010

A090010( self)

Permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A090010;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a
line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
43
sage: a.offset
1
sage: a(8)
67741129
sage: a(22)
192416593029158989003270143
sage: a.list(9)
[6, 43, 356, 3333, 34754, 398959, 4996032, 67741129, 988344062]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _repr_

_repr_( self)

sage: sloane.A090010._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n zeros not on a
line.'

Class: A090012

class A090012

A090012( self)

Permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line.

, and

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A090012;a
Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a
line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
3
sage: a(2)
9
sage: a.offset
1
sage: a(8)
890901
sage: a(22)
129020386652297208795129
sage: a.list(9)
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A090012._eval(n) for n in range(1,11)]
[3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387, 112203465]

_repr_( self)

sage: sloane.A090012._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on
a line.'

Class: A090013

class A090013

A090013( self)

Permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A090013;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a
line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
4
sage: a(2)
16
sage: a.offset
1
sage: a(8)
3481096
sage: a(22)
1112998577171142607670336
sage: a.list(9)
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A090013._eval(n) for n in range(1,11)]
[4, 16, 84, 536, 4004, 34176, 327604, 3481096, 40585284, 514872176]

_repr_( self)

sage: sloane.A090013._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on
a line.'

Class: A090014

class A090014

A090014( self)

Permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A090014;a
Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on a
line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
5
sage: a(2)
25
sage: a.offset
1
sage: a(8)
11016595
sage: a(22)
7469733600354446865509725
sage: a.list(9)
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A090014._eval(n) for n in range(1,11)]
[5, 25, 155, 1135, 9545, 90445, 952175, 11016595, 138864365, 1893369505]

_repr_( self)

sage: sloane.A090014._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=4 and n-1 zeros not on
a line.'

Class: A090015

class A090015

A090015( self)

Permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A090015;a
Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on a
line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
6
sage: a(2)
36
sage: a.offset
1
sage: a(8)
29976192
sage: a(22)
41552258517692116794936876
sage: a.list(9)
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A090015._eval(n) for n in range(1,10)]
[6, 36, 258, 2136, 19998, 208524, 2393754, 29976192, 406446774]

_repr_( self)

sage: sloane.A090015._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=3 and n-1 zeros not on
a line.'

Class: A090016

class A090016

A090016( self)

Permanent of (0,1)-matrix of size $n \times (n+d)$ with and zeros not on a line.

This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202.

REFERENCES: Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A090016;a
Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a
line.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(1)
7
sage: a(2)
49
sage: a.offset
1
sage: a(8)
72737161
sage: a(22)
199341969448774341802426289
sage: a.list(9)
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

Author: Jaap Spies (2007-01-23)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A090016._eval(n) for n in range(1,10)]
[7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191]

_repr_( self)

sage: sloane.A090016._repr_()
'Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on
a line.'

Class: A111774

class A111774

A111774( self)

Sequence of numbers of the third kind, i.e., numbers that can be written as a sum of at least three consecutive positive integers.

Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of consecutive integers (other than the trivial for ).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A111774; a
Numbers that can be written as a sum of at least three consecutive positive
integers.
sage: a(1)
6
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
141
sage: a(156)
209
sage: a(302)
386
sage: a.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2007-01-13)

Functions: is_number_of_the_third_kind, $\,$ list

is_number_of_the_third_kind( self, n)

This function returns True if and only if is a number of the third kind.

A number is of the third kind if it can be written as a sum of at least three consecutive positive integers. Odd primes can only be written as a sum of two consecutive integers. Powers of 2 do not have a representation as a sum of consecutive integers (other than the trivial for ).

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

Input:

n: - positive integer

Output:

True: - if n is not prime and not a power of 2 False -

sage: a = sloane.A111774
sage: a.is_number_of_the_third_kind(6)
True
sage: a.is_number_of_the_third_kind(100)
True
sage: a.is_number_of_the_third_kind(16)
False
sage: a.is_number_of_the_third_kind(97)
False

Author: Jaap Spies (2006-12-09)

list( self, n)

sage: sloane.A111774.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]

Special Functions: __init__, $\,$ _eval, $\,$ _precompute, $\,$ _repr_

_eval( self, n)

sage: [sloane.A111774._eval(n) for n in range(1,11)]
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A111774._b)
sage: sloane.A111774._precompute()
sage: len(sloane.A111774._b) - initial > 0
True

_repr_( self)

sage: sloane.A111774._repr_()
'Numbers that can be written as a sum of at least three consecutive
positive integers.'

Class: A111775

class A111775

A111775( self)

Number of ways can be written as a sum of at least three consecutive integers.

Powers of 2 and (odd) primes can not be written as a sum of at least three consecutive integers. strongly depends on the number of odd divisors of (A001227): Suppose is to be written as sum of consecutive integers starting with , then . Only one of the factors is odd. For each odd divisor of there is a unique corresponding , and must be excluded.

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A111775; a
Number of ways n can be written as a sum of at least three consecutive
integers.

sage: a(1)
0
sage: a(0)
0

We have a(15)=2 because 15 = 4+5+6 and 15 = 1+2+3+4+5. The number of odd divisors of 15 is 4.

sage: a(15)
2

sage: a(100)
2
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0]
sage: a(1/3)
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer

Author: Jaap Spies (2006-12-09)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A111775._eval(n) for n in range(10)]
[0, 0, 0, 0, 0, 0, 1, 0, 0, 1]

_repr_( self)

sage: sloane.A111775._repr_()
'Number of ways n can be written as a sum of at least three consecutive
integers.'

Class: A111776

class A111776

A111776( self)

The th term of the sequence is the largest such that can be written as sum of consecutive integers.

is the sum of at most consecutive positive integers. Suppose is to be written as sum of consecutive integers starting with , then . Only one of the factors is odd. For each odd divisor of there is a unique corresponding . is the largest among those . See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

Input:

n: - non negative integer

Output:

integer: - function value

sage: a = sloane.A111776; a
a(n) is the largest k such that n can be written as sum of k consecutive
integers.
sage: a(0)
1
sage: a(2)
1
sage: a.list(9)
[1, 1, 1, 2, 1, 2, 3, 2, 1]

Author: Jaap Spies (2007-01-13)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A111776._eval(n) for n in range(10)]
[1, 1, 1, 2, 1, 2, 3, 2, 1, 3]

_repr_( self)

sage: sloane.A111776._repr_()
'a(n) is the largest k such that n can be written as sum of k consecutive
integers.'

Class: A111787

class A111787

A111787( self)

This function returns the -th number of Sloane's sequence A111787

if is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose is to be written as sum of consecutive integers starting with , then . Let be the smallest odd prime divisor of then .

See: http://www.jaapspies.nl/mathfiles/problem2005-2C.pdf

Input:

n: - positive integer

Output:

integer: - function value

sage: a = sloane.A111787; a
a(n) is the least k >= 3 such that n can be written as sum of k consecutive
integers. a(n)=0 if such a k does not exist.
sage: a.offset
1
sage: a(1)
0
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(100)
5
sage: a(256)
0
sage: a(29)
0
sage: a.list(20)
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4, 0, 3, 0, 4, 3, 0, 0, 3, 0, 5]
sage: a(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be a positive integer

Author: - Jaap Spies (2007-01-14)

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A111787._eval(n) for n in range(1,11)]
[0, 0, 0, 0, 0, 3, 0, 0, 3, 4]

_repr_( self)

sage: sloane.A111787._repr_()
'a(n) is the least k >= 3 such that n can be written as sum of k
consecutive integers. a(n)=0 if such a k does not exist.'

Class: ExponentialNumbers

class ExponentialNumbers

ExponentialNumbers( self, a)

A sequence of Exponential numbers.

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0

Special Functions: __init__, $\,$ _eval, $\,$ _repr_

_eval( self, n)

sage: [sloane.A000110._eval(n) for n in range(10)]
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

_repr_( self)

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(4)._repr_()
'Sequence of Exponential numbers around 4'

Class: ExtremesOfPermanentsSequence

class ExtremesOfPermanentsSequence

Functions: gen, $\,$ list

gen( self, a0, a1, d)

sage: it = sloane.A000153.gen(0,1,2)
sage: [it.next() for i in range(5)]
[0, 1, 2, 7, 32]

list( self, n)

sage: sloane.A000153.list(8)
       [0, 1, 2, 7, 32, 181, 1214, 9403]

Special Functions: _eval, $\,$ _precompute

_eval( self, n)

sage: [sloane.A000153._eval(n) for n in range(8)]
       [0, 1, 2, 7, 32, 181, 1214, 9403]

_precompute( self, [how_many=20])

sage: sloane.A000153._precompute()
sage: v1 = len(sloane.A000153._b)
sage: sloane.A000153._precompute(10)
sage: len(sloane.A000153._b) - v1
10

Class: ExtremesOfPermanentsSequence2

class ExtremesOfPermanentsSequence2

Functions: gen

gen( self, a0, a1, d)

sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
sage: e = ExtremesOfPermanentsSequence2()
sage: it = e.gen(6,43,6)
sage: [it.next() for i in range(5)]
[6, 43, 307, 2542, 23799]

Class: RecurrenceSequence

class RecurrenceSequence

Functions: list

list( self, n)

sage: sloane.A001110.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]

Special Functions: _eval, $\,$ _precompute

_eval( self, n)

sage: [sloane.A001110._eval(n) for n in range(5)]
[0, 1, 36, 1225, 41616]

_precompute( self, [how_many=20])

sage: initial = len(sloane.A001110._b)
sage: sloane.A001110._precompute(10)
sage: len(sloane.A001110._b) - initial == 10
True

Class: RecurrenceSequence2

class RecurrenceSequence2

Functions: list

list( self, n)

sage: sloane.A001906.list(10)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

Special Functions: _eval, $\,$ _precompute

_eval( self, n)

sage: [sloane.A001906._eval(n) for n in range(10)]
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]

_precompute( self, [how_many=150])

sage: initial = len(sloane.A001906._b)
sage: sloane.A001906._precompute(10)
sage: len(sloane.A001906._b) - initial == 10
True

Class: Sloane

class Sloane

A collection of Sloane generating functions.

This class inspects sage.combinat.sloane_functions, accumulating all the SloaneSequence classes starting with 'A'. These are listed for tab completion, but not instantiated until requested.

Ensure we have lots of entries:

sage: len(sloane.trait_names()) > 100
True

And ensure none are being incorrectly returned:

sage: [ None for n in sloane.trait_names() if not n.startswith('A') ]
[]

Ensure we can access dynamic constructions and cache correctly:

sage: s = sloane.A000587
sage: s is sloane.A000587
True

And that we can access other functions in parent classes:

sage: sloane.__class__
<class 'sage.combinat.sloane_functions.Sloane'>

Author: Nick Alexander

Functions: trait_names

trait_names( self)

List Sloane generating functions for tab-completion.

The member classes are inspected from module sage.combinat.sloane_functions.

They must be sub classes of SloaneSequence and must start with 'A'. These restrictions are only to prevent typos, incorrect inspecting, etc.

sage: type(sloane.trait_names())
<type 'list'>

Special Functions: __getattribute__

__getattribute__( self, name)

Construct and cache unique instances of Sloane generating function objects .

sage: sloane.__getattribute__('A000001')
Number of groups of order n.
sage: sloane.__getattribute__('dog')
Traceback (most recent call last):
...
AttributeError: dog

Class: SloaneSequence

class SloaneSequence

Base class for a Slone integer sequence.

We create a dummy sequence:

SloaneSequence( self, [offset=1])

A sequence starting at offset (=1 by default).

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence().offset
1
sage: SloaneSequence(4).offset
4

Functions: list

list( self, n)

Return n terms of the sequence: sequence[offset], sequence[offset+1], ... , sequence[offset+n-1].

sage: sloane.A000012.list(4)
[1, 1, 1, 1]

Special Functions: __call__, $\,$ __cmp__, $\,$ __getitem__, $\,$ __init__, $\,$ __iter__, $\,$ _eval, $\,$ _repr_

__call__( self, n)

sage: sloane.A000007(2)
0
sage: sloane.A000007('a')
Traceback (most recent call last):
...
TypeError: input must be an int, long, or Integer
sage: sloane.A000007(-1)
Traceback (most recent call last):
...
ValueError: input n (=-1) must be an integer >= 0
sage: sloane.A000001(0)
Traceback (most recent call last):            
...
ValueError: input n (=0) must be a positive integer

__cmp__( self, other)

sage: cmp(sloane.A000007,sloane.A000045) == 0
False
sage: cmp(sloane.A000007,sloane.A000007) == 0
True

__getitem__( self, n)

Return sequence[n].

We interpret slices as best we can, but our sequences are infinite so we want to prevent some mis-incantations.

Therefore, we abitrarily cap slices to be at most LENGTH=100000 elements long. Since many Sloane sequences are costly to compute, this is probably not an unreasonable decision, but just in case, list does not cap length.

sage: sloane.A000012[3]
1
sage: sloane.A000012[:4]
[1, 1, 1, 1]
sage: sloane.A000012[:10]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: sloane.A000012[4:10]
[1, 1, 1, 1, 1, 1]
sage: sloane.A000012[0:1000000000]
Traceback (most recent call last):
...
IndexError: slice (=slice(0, 1000000000, None)) too long

__iter__( self)

sage: iter(sloane.A000012)
Traceback (most recent call last):
...
NotImplementedError

_eval( self, n)

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence(0)._eval(4)
Traceback (most recent call last):
...
NotImplementedError

_repr_( self)

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence(4)._repr_()
Traceback (most recent call last):
...
NotImplementedError

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