Functions that compute some of the sequences in Sloane’s tables¶

EXAMPLES:

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
True

We agree with the online database::

sage: for t in sloane.trait_names():    # long time; optional -- internet
....:     online_list = list(oeis(t).first_terms())
....:     L = max(2, len(online_list) // 2)
....:     sage_list = sloane.__getattribute__(t).list(L)
....:     if online_list[:L] != sage_list:
....:         print t, 'seems wrong'

• If you want to get more informations relative to a sequence (references, links, examples, programs, ...), you can use the On-Line Encyclopedia of Integer Sequences provided by the OEIS module.
• If you plan to do a lot of automatic searches for subsequences, you should consider installing SloaneEncyclopedia, a local partial copy of the OEIS.

AUTHORS:

• William Stein: framework
• Jaap Spies: most sequences
• Nick Alexander: updated framework
class sage.combinat.sloane_functions.A000001

Number of groups of order $$n$$.

Note: The database_gap-4.4.9 must be installed for $$n > 50$$.

run sage -i database_gap-4.4.9 or higher first.

INPUT:

• n - positive integer

OUTPUT: integer

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-02-04)
class sage.combinat.sloane_functions.A000004

The zero sequence.

INPUT:

• n - non negative integer

OUTPUT:

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2006-12-10)
class sage.combinat.sloane_functions.A000005

The sequence $$tau(n)$$, which is the number of divisors of $$n$$.

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

INPUT:

• n - positive integer

OUTPUT:

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2006-12-10)
• William Stein (2007-01-08)
class sage.combinat.sloane_functions.A000007

The characteristic function of 0: $$a(n) = 0^n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-12)
class sage.combinat.sloane_functions.A000008

Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

sage: a = sloane.A000008;a
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
sage: a(0)
1
sage: a(1)
1
sage: a(13)
16
sage: a.list(14)
[1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16]

AUTHOR:

class sage.combinat.sloane_functions.A000009

Number of partitions of $$n$$ into odd parts.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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)
cf()

EXAMPLES:

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(n)

EXAMPLES:

sage: sloane.A000009.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
class sage.combinat.sloane_functions.A000010

The integer sequence A000010 is Euler’s totient function.

Number of positive integers $$i < n$$ that are relative prime to $$n$$. Number of totatives of $$n$$.

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-12)
class sage.combinat.sloane_functions.A000012

The all 1’s sequence.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-12)
class sage.combinat.sloane_functions.A000015

Smallest prime power $$\geq n$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000016

Sloane’s A000016

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000027

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.

EXAMPLES:

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]
class sage.combinat.sloane_functions.A000030

Initial digit of $$n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000032

Lucas numbers (beginning at 2): $$L(n) = L(n-1) + L(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000035

A simple periodic sequence.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
class sage.combinat.sloane_functions.A000040

The prime numbers.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-17)
class sage.combinat.sloane_functions.A000041

$$a(n)$$ = number of partitions of $$n$$ (the partition numbers).

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000043

Primes $$p$$ such that $$2^p - 1$$ is prime. $$2^p - 1$$ is then called a Mersenne prime.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000045

Sequence of Fibonacci numbers, offset 0,4.

REFERENCES:

We have one more. Our first Fibonacci number is 0.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-13)
fib()

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

EXAMPLES:

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

EXAMPLES:

sage: sloane.A000045.list(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
class sage.combinat.sloane_functions.A000069

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
class sage.combinat.sloane_functions.A000073

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
list(n)

EXAMPLES:

sage: sloane.A000073.list(10)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
class sage.combinat.sloane_functions.A000079

Powers of 2: $$a(n) = 2^n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000085

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-03)
class sage.combinat.sloane_functions.A000100

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000108

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-12)
class sage.combinat.sloane_functions.A000110

The sequence of Bell numbers.

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

Let $$S(n, k)$$ denote the Stirling number of the second kind. Then

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

INPUT:

• n - integer = 0

OUTPUT:

• integer - $$B_n$$

EXAMPLES:

sage: a = sloane.A000110; a
Sequence of Bell numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
sage: a.list(10)
[1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147]

AUTHORS:

• Nick Alexander
class sage.combinat.sloane_functions.A000120

1’s-counting sequence: number of 1’s in binary expansion of $$n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
f(n)

EXAMPLES:

sage: [sloane.A000120.f(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
class sage.combinat.sloane_functions.A000124

Central polygonal numbers (the Lazy Caterer’s sequence): $$n(n+1)/2 + 1$$.

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000129

Pell numbers: $$a(0) = 0$$, $$a(1) = 1$$; for $$n > 1$$, $$a(n) = 2a(n-1) + a(n-2)$$.

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000142

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

Order of symmetric group $$S_n$$, number of permutations of $$n$$ letters.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-12)
class sage.combinat.sloane_functions.A000153

$$a(n) = n*a(n-1) + (n-2)*a(n-2)$$, with $$a(0) = 0$$, $$a(1) = 1$$.

With offset 1, permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=2$$ and $$n$$ 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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A000165

Double factorial numbers: $$(2n)!! = 2^n*n!$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-24)
class sage.combinat.sloane_functions.A000166

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

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A000169

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000203

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

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A000204

Lucas numbers (beginning with 1): $$L(n) = L(n-1) + L(n-2)$$ with $$L(1) = 1$$, $$L(2) = 3$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A000213

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
list(n)

EXAMPLES:

sage: sloane.A000213.list(10)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
class sage.combinat.sloane_functions.A000217

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000225

$$2^n - 1$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000244

Powers of 3: $$a(n) = 3^n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000255

$$a(n) = n*a(n-1) + (n-1)*a(n-2)$$, with $$a(0) = 1$$, $$a(1) = 1$$.

With offset 1, permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=1$$ and $$n$$ 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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A000261

$$a(n) = n*a(n-1) + (n-3)*a(n-2)$$, with $$a(1) = 1$$, $$a(2) = 1$$.

With offset 1, permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=3$$ and $$n$$ 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.

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A000272

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

INPUT:

• n - integer

OUTPUT:

• integer - function value

EXAMPLES:

sage: a = sloane.A000272;a
Number of labeled rooted trees with n nodes: n^(n-2).
sage: a(0)
1
sage: a(1)
1
sage: a(2)
1
sage: a(10)
100000000
sage: a.list(12)
[1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000290

The squares: $$a(n) = n^2$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000292

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000302

Powers of 4: $$a(n) = 4^n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000312

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000326

Pentagonal numbers: $$n(3n-1)/2$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000330

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000396

Perfect numbers: equal to sum of proper divisors.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000578

The cubes: $$a(n) = n^3$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A000583

Fourth powers: $$a(n) = n^4$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-04)
class sage.combinat.sloane_functions.A000587

The sequence of Uppuluri-Carpenter numbers.

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

Let $$S(n, k)$$ denote the Stirling number of the second kind. Then

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

INPUT:

• n - integer = 0

OUTPUT:

• integer - $$C_n$$

EXAMPLES:

sage: a = sloane.A000587; a
Sequence of Uppuluri-Carpenter numbers
sage: a.offset
0
sage: a(0)
1
sage: a(100)
397577026456518507969762382254187048845620355238545130875069912944235105204434466095862371032124545552161
sage: a.list(10)
[1, -1, 0, 1, 1, -2, -9, -9, 50, 267]

AUTHORS:

• Nick Alexander
class sage.combinat.sloane_functions.A000668

Mersenne primes (of form $$2^p - 1$$ where $$p$$ is a prime).

(See A000043 for the values of $$p$$.)

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000670

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-03)
class sage.combinat.sloane_functions.A000720

$$pi(n)$$, the number of primes $$\le n$$. Sometimes called $$PrimePi(n)$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A000796

Decimal expansion of $$\pi$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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)
list(n)

EXAMPLES:

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

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

EXAMPLES:

sage: it = sloane.A000796.pi()
sage: [it.next() for i in range(10)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
class sage.combinat.sloane_functions.A000961

Prime powers

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
list(n)

EXAMPLES:

sage: sloane.A000961.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
class sage.combinat.sloane_functions.A000984

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A001006

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
class sage.combinat.sloane_functions.A001045

Jacobsthal sequence: $$a(n) = a(n-1) + 2a(n-2)$$, $$a(0) = 0$$ and $$a(1) = 1$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A001055

Number of ways of factoring $$n$$ with all factors 1.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-04)
nwf(n, m)

EXAMPLES:

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
class sage.combinat.sloane_functions.A001109

$$a(n)^2$$ is a triangular number: $$a(n) = 6*a(n-1) - a(n-2)$$ with $$a(0)=0$$, $$a(1)=1$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-24)
class sage.combinat.sloane_functions.A001110

Numbers that are both triangular and square: $$a(n) = 34a(n-1) - a(n-2) + 2$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
g(k)

EXAMPLES:

sage: sloane.A001110.g(2)
2
sage: sloane.A001110.g(1)
0
class sage.combinat.sloane_functions.A001147

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-24)
class sage.combinat.sloane_functions.A001157

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

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A001189

Number of degree-n permutations of order exactly 2.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-03)
class sage.combinat.sloane_functions.A001221

Number of different prime divisors of $$n$$

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A001222

Number of prime divisors of $$n$$ (counted with multiplicity).

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A001227

Number of odd divisors of $$n$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-14)
class sage.combinat.sloane_functions.A001333

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-01)
class sage.combinat.sloane_functions.A001358

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
list(n)

EXAMPLES:

sage: sloane.A001358.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]
class sage.combinat.sloane_functions.A001405

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A001477

The nonnegative integers.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A001694

This function returns the $$n$$-th Powerful Number:

A positive integer $$n$$ is powerful if for every prime $$p$$ dividing $$n$$, $$p^2$$ also divides $$n$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-14)
is_powerful(n)

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

A positive integer $$n$$ is powerful if for every prime $$p$$ dividing $$n$$, $$p^2$$ also divides $$n$$. See Sloane’s OEIS A001694.

INPUT:

• n - integer

OUTPUT:

• True - if $$n$$ is a Powerful number, else False

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2006-12-07)
list(n)

EXAMPLES:

sage: sloane.A001694.list(9)
[1, 4, 8, 9, 16, 25, 27, 32, 36]
class sage.combinat.sloane_functions.A001836

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

Euler’s totient function is also known as euler_phi, euler_phi is a standard Sage function.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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 Euler’s totient function A000010. [53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893]

AUTHORS:

• Jaap Spies (2007-01-17)
list(n)

EXAMPLES:

sage: sloane.A001836.list(9)
[53, 83, 158, 263, 293, 368, 578, 683, 743]
class sage.combinat.sloane_functions.A001906

$$F(2n) =$$ bisection of Fibonacci sequence: $$a(n)=3a(n-1)-a(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A001909

$$a(n) = n*a(n-1) + (n-4)*a(n-2)$$, with $$a(2) = 0$$, $$a(3) = 1$$.

With offset 1, permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=4$$ and $$n$$ 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.

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A001910

$$a(n) = n*a(n-1) + (n-5)*a(n-2)$$, with $$a(3) = 0$$, $$a(4) = 1$$.

With offset 1, permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=5$$ and $$n$$ 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.

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A001969

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
class sage.combinat.sloane_functions.A002110

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A002113

Palindromes in base 10.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
list(n)

EXAMPLES:

sage: sloane.A002113.list(15)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]
class sage.combinat.sloane_functions.A002275

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A002378

Oblong (or pronic, or heteromecic) numbers: $$n(n+1)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A002620

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A002808

The composite numbers: numbers $$n$$ of the form $$xy$$ for $$x > 1$$ and $$y > 1$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
list(n)

EXAMPLES:

sage: sloane.A002808.list(10)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
class sage.combinat.sloane_functions.A003418

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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)
class sage.combinat.sloane_functions.A004086

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
class sage.combinat.sloane_functions.A004526

The nonnegative integers repeated

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A005100

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
list(n)

EXAMPLES:

sage: sloane.A005100.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]
class sage.combinat.sloane_functions.A005101

Abundant numbers (sum of divisors of $$n$$ exceeds $$2n$$).

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
list(n)

EXAMPLES:

sage: sloane.A005101.list(10)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
class sage.combinat.sloane_functions.A005117

Square-free numbers

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
list(n)

EXAMPLES:

sage: sloane.A005117.list(10)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
class sage.combinat.sloane_functions.A005408

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-26)
class sage.combinat.sloane_functions.A005843

The even numbers: $$a(n) = 2n$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-03)
class sage.combinat.sloane_functions.A006318

Large Schroeder numbers.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-03)
class sage.combinat.sloane_functions.A006530

Largest prime dividing $$n$$ (with $$a(1)=1$$).

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
class sage.combinat.sloane_functions.A006882

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-24)
df()

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

EXAMPLES:

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

EXAMPLES:

sage: sloane.A006882.list(10)
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
class sage.combinat.sloane_functions.A007318

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

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-31)
class sage.combinat.sloane_functions.A008275

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

The unsigned numbers are also called Stirling cycle numbers:

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
s(n, k)

EXAMPLES:

sage: sloane.A008275.s(4,2)
11
sage: sloane.A008275.s(5,2)
-50
sage: sloane.A008275.s(5,3)
35
class sage.combinat.sloane_functions.A008277

Triangle of Stirling numbers of 2nd kind, $$S2(n,k)$$, $$n \ge 1$$, $$1 \le k \le n$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-31)
s2(n, k)

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

EXAMPLES:

sage: sloane.A008277.s2(4,2)
7
class sage.combinat.sloane_functions.A008683

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A010060

Thue-Morse sequence.

Let $$A_k$$ denote the first $$2^k$$ terms; then $$A_0 = 0$$, and for $$k \ge 0$$, $$A_{k+1} = A_k B_k$$, where $$B_k$$ is obtained from $$A_k$$ by interchanging 0’s and 1’s.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-02)
class sage.combinat.sloane_functions.A015521

Linear 2nd order recurrence, $$a(0)=0$$, $$a(1)=1$$ and $$a(n) = 3 a(n-1) + 4 a(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A015523

Linear 2nd order recurrence, $$a(0)=0$$, $$a(1)=1$$ and $$a(n) = 3 a(n-1) + 5 a(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A015530

Linear 2nd order recurrence, $$a(0)=0$$, $$a(1)=1$$ and $$a(n) = 4 a(n-1) + 3 a(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A015531

Linear 2nd order recurrence, $$a(0)=0$$, $$a(1)=1$$ and $$a(n) = 4 a(n-1) + 5 a(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A015551

Linear 2nd order recurrence, $$a(0)=0$$, $$a(1)=1$$ and $$a(n) = 6 a(n-1) + 5 a(n-2)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A018252

The nonprime numbers, starting with 1.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-04)
class sage.combinat.sloane_functions.A020639

Least prime dividing $$n$$ with $$a(1)=1$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-25)
list(n)

EXAMPLES:

sage: sloane.A020639.list(10)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
class sage.combinat.sloane_functions.A046660(offset=1)

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

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

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A049310

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

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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']

AUTHORS:

• Jaap Spies (2007-01-31)
class sage.combinat.sloane_functions.A051959

Linear second order recurrence. A051959.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
g(k)

EXAMPLES:

sage: sloane.A051959.g(2)
15
sage: sloane.A051959.g(1)
0
class sage.combinat.sloane_functions.A055790

$$a(n) = n*a(n-1) + (n-2)*a(n-2) [a(0) = 0, a(1) = 2]$$.

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A061084

Fibonacci-type sequence based on subtraction: $$a(0) = 1$$, $$a(1) = 2$$ and $$a(n) = a(n-2)-a(n-1)$$.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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']

AUTHORS:

• Jaap Spies (2007-01-18)
class sage.combinat.sloane_functions.A064553

$$a(1) = 1$$, $$a(prime(i)) = i + 1$$ for $$i > 0$$ and $$a(u \cdot v) = a(u) \cdot a(v)$$ for $$u, v > 0$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-02-04)
class sage.combinat.sloane_functions.A079922(offset=1)

function returns solutions to the Dancing School problem with $$n$$ girls and $$n+3$$ boys.

The value is $$per(B)$$, the permanent of the (0,1)-matrix $$B$$ of size $$n \times n+3$$ with $$b(i,j)=1$$ 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

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-14)
class sage.combinat.sloane_functions.A079923(offset=1)

function returns solutions to the Dancing School problem with $$n$$ girls and $$n+4$$ boys.

The value is $$per(B)$$, the permanent of the (0,1)-matrix $$B$$ of size $$n \times n+3$$ with $$b(i,j)=1$$ 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

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-17)
class sage.combinat.sloane_functions.A082411

Second-order linear recurrence sequence with $$a(n) = a(n-1) + a(n-2)$$.

$$a(0) = 407389224418$$, $$a(1) = 76343678551$$. This is the second-order linear recurrence sequence with $$a(0)$$ and $$a(1)$$ co-prime, that R. L. Graham in 1964 stated did not contain any primes.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A083103

Second-order linear recurrence sequence with $$a(n) = a(n-1) + a(n-2)$$.

$$a(0) = 1786772701928802632268715130455793$$, $$a(1) = 1059683225053915111058165141686995$$. This is the second-order linear recurrence sequence with $$a(0)$$ and $$a(1)$$ 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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A083104

Second-order linear recurrence sequence with $$a(n) = a(n-1) + a(n-2)$$.

$$a(0) = 331635635998274737472200656430763$$, $$a(1) = 1510028911088401971189590305498785$$. This is the second-order linear recurrence sequence with $$a(0)$$ and $$a(1)$$ co-prime. It was found by Ronald Graham in 1990.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A083105

Second-order linear recurrence sequence with $$a(n) = a(n-1) + a(n-2)$$.

$$a(0) = 62638280004239857$$, $$a(1) = 49463435743205655$$. This is the second-order linear recurrence sequence with $$a(0)$$ and $$a(1)$$ co-prime. It was found by Donald Knuth in 1990.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A083216

Second-order linear recurrence sequence with $$a(n) = a(n-1) + a(n-2)$$.

$$a(0) = 20615674205555510$$, $$a(1) = 3794765361567513$$. This is a second-order linear recurrence sequence with $$a(0)$$ and $$a(1)$$ 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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-19)
class sage.combinat.sloane_functions.A090010

Permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=6$$ and $$n$$ zeros not on a line.

 a(n) = (n+5)*a(n-1) + (n-1)*a(n-2), a(1)=6, a(2)=43.

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A090012

Permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=2$$ and $$n-1$$ zeros not on a line.

$$a(n) = (n+1)*a(n-1) + (n-2)*a(n-2)$$, $$a(1)=3$$ and $$a(2)=9$$

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A090013

Permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=3$$ and $$n-1$$ zeros not on a line.

$$a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=4, a(2)=16]$$

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A090014

Permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=4$$ and $$n-1$$ zeros not on a line.

$$a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=5, a(2)=25]$$

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A090015

Permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=5$$ and $$n-1$$ zeros not on a line.

$$a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=6, a(2)=36]$$

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A090016

Permanent of (0,1)-matrix of size $$n \times (n+d)$$ with $$d=6$$ and $$n-1$$ zeros not on a line.

$$a(n) = (n+1)*a(n-1) + (n-2)*a(n-2) [a(1)=7, a(2)=49]$$

$$A090016 a(n) = A090010(n-1) + A090010(n), a(1)=7$$

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

EXAMPLES:

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]

AUTHORS:

• Jaap Spies (2007-01-23)
class sage.combinat.sloane_functions.A109814

The $$n$$ th term of the sequence $$a(n)$$ is the largest $$k$$ such that $$n$$ can be written as sum of $$k$$ consecutive integers.

By definition, $$n$$ is the sum of at most $$a(n)$$ consecutive positive integers. Suppose $$n$$ is to be written as sum of $$k$$ consecutive integers starting with $$m$$, then $$2n = k(2m + k - 1)$$. Only one of the factors is odd. For each odd divisor $$d$$ of $$n$$ there is a unique corresponding $$k = min(d,2n/d)$$. $$a(n)$$ can be alternatively defined as the largest among those $$k$$ .

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

sage: a = sloane.A109814; a
a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
sage: a(0)
Traceback (most recent call last):
...
ValueError: input n (=0) must be a positive integer
sage: a(2)
1
sage: a.list(9)
[1, 1, 2, 1, 2, 3, 2, 1, 3]

AUTHORS:

• Jaap Spies (2007-01-13)
class sage.combinat.sloane_functions.A111774

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 $$k$$ consecutive integers (other than the trivial $$n = n$$ for $$k = 1$$).

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-13)
is_number_of_the_third_kind(n)

This function returns True if and only if $$n$$ 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 $$k$$ consecutive integers (other than the trivial $$n = n$$ for $$k = 1$$).

INPUT:

• n - positive integer

OUTPUT:

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

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2006-12-09)
list(n)

EXAMPLES:

sage: sloane.A111774.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
class sage.combinat.sloane_functions.A111775

Number of ways $$n$$ 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. $$a(n)$$ strongly depends on the number of odd divisors of $$n$$ (A001227): Suppose $$n$$ is to be written as sum of $$k$$ consecutive integers starting with $$m$$, then $$2n = k(2m + k - 1)$$. Only one of the factors is odd. For each odd divisor of $$n$$ there is a unique corresponding $$k$$, $$k=1$$ and $$k=2$$ must be excluded.

INPUT:

• n - non negative integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2006-12-09)
class sage.combinat.sloane_functions.A111787

This function returns the $$n$$-th number of Sloane’s sequence A111787

$$a(n)=0$$ if $$n$$ is an odd prime or a power of 2. For numbers of the third kind (see A111774) we proceed as follows: suppose $$n$$ is to be written as sum of $$k$$ consecutive integers starting with $$m$$, then $$2n = k(2m + k - 1)$$. Let $$p$$ be the smallest odd prime divisor of $$n$$ then $$a(n) = min(p,2n/p)$$.

INPUT:

• n - positive integer

OUTPUT:

• integer - function value

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2007-01-14)
class sage.combinat.sloane_functions.ExponentialNumbers(a)

A sequence of Exponential numbers.

EXAMPLES:

sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0
class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence(offset=1)

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

EXAMPLES:

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence().offset
1
sage: SloaneSequence(4).offset
4
gen(a0, a1, d)

EXAMPLES:

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

EXAMPLES:

sage: sloane.A000153.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]
class sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2(offset=1)

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

EXAMPLES:

sage: from sage.combinat.sloane_functions import SloaneSequence
sage: SloaneSequence().offset
1
sage: SloaneSequence(4).offset
4
gen(a0, a1, d)

EXAMPLES:

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 sage.combinat.sloane_functions.RecurrenceSequence(offset=1)

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

EXAMPLES:

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

EXAMPLES:

sage: sloane.A001110.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]
class sage.combinat.sloane_functions.RecurrenceSequence2(offset=1)

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

EXAMPLES:

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

EXAMPLES:

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

EXAMPLES: 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'>

AUTHORS:

• Nick Alexander
trait_names()

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.

EXAMPLES:

sage: type(sloane.trait_names())
<type 'list'>
class sage.combinat.sloane_functions.SloaneSequence(offset=1)

Base class for a Sloane integer sequence.

EXAMPLES:

We create a dummy sequence:

list(n)

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

sage: sloane.A000012.list(4)
[1, 1, 1, 1]
sage.combinat.sloane_functions.perm_mh(m, h)

This functions calculates $$f(g,h)$$ from Sloane’s sequences A079908-A079928

INPUT:

• m - positive integer
• h - non negative integer

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

EXAMPLES:

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

AUTHORS:

• Jaap Spies (2006)
sage.combinat.sloane_functions.recur_gen2(a0, a1, a2, a3)

homogeneous 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)

EXAMPLES:

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]
sage.combinat.sloane_functions.recur_gen2b(a0, a1, a2, a3, b)

inhomogenous second-order linear recurrence generator with fixed coefficients and $$b = f(n)$$

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

EXAMPLES:

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]
sage.combinat.sloane_functions.recur_gen3(a0, a1, a2, a3, a4, a5)

homogeneous 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)

EXAMPLES:

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]

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