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
sage: a == loads(dumps(a))
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'
See also
AUTHORS:
Bases: sage.combinat.sloane_functions.SloaneSequence
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:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The zero sequence.
INPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The sequence , which is the number of divisors of
.
This sequence is also denoted (also called
or
), the number of
divisors of n.
INPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The characteristic function of 0: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of partitions of into odd parts.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: it = sloane.A000009.cf()
sage: [next(it) for i in range(14)]
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
EXAMPLES:
sage: sloane.A000009.list(14)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18]
Bases: sage.combinat.sloane_functions.SloaneSequence
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 : count numbers
and prime to
. euler_phi is a standard Sage function
implemented in PARI
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The all 1’s sequence.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Smallest prime power .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Sloane’s A000016
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
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]
Bases: sage.combinat.sloane_functions.SloaneSequence
Initial digit of .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Lucas numbers (beginning at 2): .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
A simple periodic sequence.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The prime numbers.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
= number of partitions of
(the partition
numbers).
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Primes such that
is prime.
is then called a Mersenne prime.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Sequence of Fibonacci numbers, offset 0,4.
REFERENCES:
We have one more. Our first Fibonacci number is 0.
INPUT:
OUTPUT:
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:
Returns a generator over all Fibonacci numbers, starting with 0.
EXAMPLES:
sage: it = sloane.A000045.fib()
sage: [next(it) for i in range(10)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
EXAMPLES:
sage: sloane.A000045.list(10)
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34]
Bases: sage.combinat.sloane_functions.SloaneSequence
Odious numbers: odd number of 1’s in binary expansion.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 0, 0, 1, ...
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000073.list(10)
[0, 0, 1, 1, 2, 4, 7, 13, 24, 44]
Bases: sage.combinat.sloane_functions.SloaneSequence
Powers of 2: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of self-inverse permutations on letters, also
known as involutions; number of Young tableaux with
cells.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Catalan numbers:
.
Also called Segner numbers.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExponentialNumbers
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
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
1’s-counting sequence: number of 1’s in binary expansion of
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: [sloane.A000120.f(n) for n in range(10)]
[0, 1, 1, 2, 1, 2, 2, 3, 1, 2]
Bases: sage.combinat.sloane_functions.SloaneSequence
Central polygonal numbers (the Lazy Caterer’s sequence):
.
Or, maximal number of pieces formed when slicing a pancake with
cuts.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Pell numbers: ,
; for
,
.
Denominators of continued fraction convergents to
.
See also A001333
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Factorial numbers:
Order of symmetric group , number of permutations of
letters.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Double factorial numbers: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of labeled rooted trees with nodes:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The sequence , where
is the
sum of the divisors of
. Also called
.
The function sigma(n, k) implements
in Sage.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Lucas numbers (beginning with 1):
with
,
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3). Starting with 1, 1, 1, ...
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000213.list(10)
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]
Bases: sage.combinat.sloane_functions.SloaneSequence
Triangular numbers: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Powers of 3: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
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.
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of labeled rooted trees on nodes:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The squares: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Tetrahedral (or pyramidal) numbers:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Powers of 4: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of labeled mappings from points to themselves
(endofunctions):
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Pentagonal numbers: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Square pyramidal numbers”
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Perfect numbers: equal to sum of proper divisors.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The cubes: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Fourth powers: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExponentialNumbers
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
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Mersenne primes (of form where
is a
prime).
(See A000043 for the values of .)
Warning: a(39) has 4,053,946 digits!
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of preferential arrangements of labeled
elements; or number of weak orders on
labeled
elements.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
, the number of primes
. Sometimes
called
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Decimal expansion of .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000796.list(10)
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
Based on an algorithm of Lambert Meertens The ABC-programming language!!!
EXAMPLES:
sage: it = sloane.A000796.pi()
sage: [next(it) for i in range(10)]
[3, 1, 4, 1, 5, 9, 2, 6, 5, 3]
Bases: sage.combinat.sloane_functions.SloaneSequence
Prime powers
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A000961.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 11, 13]
Bases: sage.combinat.sloane_functions.SloaneSequence
Central binomial coefficients:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Motzkin numbers: number of ways of drawing any number of
nonintersecting chords among points on a circle.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Jacobsthal sequence: ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of ways of factoring with all factors 1.
INPUT:
OUTPUT:
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:
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
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
is a triangular number:
with
,
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence
Numbers that are both triangular and square:
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001110.g(2)
2
sage: sloane.A001110.g(1)
0
Bases: sage.combinat.sloane_functions.SloaneSequence
Double factorial numbers:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The sequence , sum of squares of divisors of
.
The function sigma(n, k) implements in Sage.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of degree-n permutations of order exactly 2.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of different prime divisors of
Also called omega(n) or . Maximal number of
terms in any factorization of
. Number of prime powers
that divide
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of prime divisors of (counted with
multiplicity).
Also called bigomega(n) or . Maximal number of
terms in any factorization of
. Number of prime powers
that divide
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Number of odd divisors of .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Numerators of continued fraction convergents to .
See also A000129
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Products of two primes.
These numbers have been called semiprimes (or semi-primes), biprimes or 2-almost primes.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001358.list(9)
[4, 6, 9, 10, 14, 15, 21, 22, 25]
Bases: sage.combinat.sloane_functions.SloaneSequence
Central binomial coefficients:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The nonnegative integers.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
This function returns the -th Powerful Number:
A positive integer is powerful if for every prime
dividing
,
also divides
.
INPUT:
OUTPUT:
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:
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:
OUTPUT:
EXAMPLES:
sage: a = sloane.A001694
sage: a.is_powerful(2500)
True
sage: a.is_powerful(20)
False
AUTHORS:
EXAMPLES:
sage: sloane.A001694.list(9)
[1, 4, 8, 9, 16, 25, 27, 32, 36]
Bases: sage.combinat.sloane_functions.SloaneSequence
Numbers such that
,
where
is Euler’s totient function.
Euler’s totient function is also known as euler_phi, euler_phi is a standard Sage function.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A001836.list(9)
[53, 83, 158, 263, 293, 368, 578, 683, 743]
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
bisection of Fibonacci sequence:
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
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.
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence
, with
,
.
With offset 1, permanent of (0,1)-matrix of size
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.
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Evil numbers: even number of 1’s in binary expansion.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Primorial numbers (first definition): product of first
primes. Sometimes written
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Palindromes in base 10.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A002113.list(15)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55]
Bases: sage.combinat.sloane_functions.SloaneSequence
Repunits: . Often denoted by
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Oblong (or pronic, or heteromecic) numbers: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently,
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The composite numbers: numbers of the form
for
and
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A002808.list(10)
[4, 6, 8, 9, 10, 12, 14, 15, 16, 18]
Bases: sage.combinat.sloane_functions.SloaneSequence
Least common multiple (or lcm) of .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Read n backwards (referred to as in many
sequences).
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The nonnegative integers repeated
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Deficient numbers: .
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A005100.list(10)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11]
Bases: sage.combinat.sloane_functions.SloaneSequence
Abundant numbers (sum of divisors of exceeds
).
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A005101.list(10)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]
Bases: sage.combinat.sloane_functions.SloaneSequence
Square-free numbers
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A005117.list(10)
[1, 2, 3, 5, 6, 7, 10, 11, 13, 14]
Bases: sage.combinat.sloane_functions.SloaneSequence
The odd numbers a(n) = 2n + 1.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The even numbers: .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Large Schroeder numbers.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Largest prime dividing (with
).
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Double factorials :
.
INPUT:
OUTPUT:
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:
Double factorials n!!: a(n)=n*a(n-2).
EXAMPLES:
sage: it = sloane.A006882.df()
sage: [next(it) for i in range(10)]
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
EXAMPLES:
sage: sloane.A006882.list(10)
[1, 1, 2, 3, 8, 15, 48, 105, 384, 945]
Bases: sage.combinat.sloane_functions.SloaneSequence
Pascal’s triangle read by rows:
,
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Triangle of Stirling numbers of first kind, ,
,
.
The unsigned numbers are also called Stirling cycle numbers:
= number of permutations of
objects
with exactly
cycles.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A008275.s(4,2)
11
sage: sloane.A008275.s(5,2)
-50
sage: sloane.A008275.s(5,3)
35
Bases: sage.combinat.sloane_functions.SloaneSequence
Triangle of Stirling numbers of 2nd kind, ,
,
.
INPUT:
OUTPUT:
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:
Returns the Stirling number S2(n,k) of the 2nd kind.
EXAMPLES:
sage: sloane.A008277.s2(4,2)
7
Bases: sage.combinat.sloane_functions.SloaneSequence
Moebius function .
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Thue-Morse sequence.
Let denote the first
terms; then
, and for
,
, where
is obtained
from
by interchanging 0’s and 1’s.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
Linear 2nd order recurrence, ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The nonprime numbers, starting with 1.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Least prime dividing with
.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A020639.list(10)
[1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
Bases: sage.combinat.sloane_functions.SloaneSequence
Excess of = number of prime divisors (with
multiplicity) - number of prime divisors (without multiplicity).
.
INPUT:
OUTPUT:
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)
Bases: sage.combinat.sloane_functions.SloaneSequence
Triangle of coefficients of Chebyshev’s :
polynomials (exponents in increasing
order).
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence
Linear second order recurrence. A051959.
INPUT:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A051959.g(2)
15
sage: sloane.A051959.g(1)
0
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2
.
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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Fibonacci-type sequence based on subtraction: ,
and
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
,
for
and
for
.
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
function returns solutions to the Dancing School problem with
girls and
boys.
The value is , the permanent of the (0,1)-matrix
of size
with
if and only if
.
REFERENCES:
INPUT:
OUTPUT:
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)
Bases: sage.combinat.sloane_functions.SloaneSequence
function returns solutions to the Dancing School problem with
girls and
boys.
The value is , the permanent of the (0,1)-matrix
of size
with
if and only if
.
REFERENCES:
INPUT:
OUTPUT:
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)
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.RecurrenceSequence2
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence2
Permanent of (0,1)-matrix of size with
and
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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Permanent of (0,1)-matrix of size 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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Permanent of (0,1)-matrix of size 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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Permanent of (0,1)-matrix of size 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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Permanent of (0,1)-matrix of size 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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
Permanent of (0,1)-matrix of size 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:
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
The th term of the sequence
is the
largest
such that
can be written as sum of
consecutive integers.
By definition, 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
.
can be alternatively defined as the
largest among those
.
See also
INPUT:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
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:
OUTPUT:
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:
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:
OUTPUT:
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:
EXAMPLES:
sage: sloane.A111774.list(12)
[6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25]
Bases: sage.combinat.sloane_functions.SloaneSequence
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
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:
OUTPUT:
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:
Bases: sage.combinat.sloane_functions.SloaneSequence
A sequence of Exponential numbers.
EXAMPLES:
sage: from sage.combinat.sloane_functions import ExponentialNumbers
sage: ExponentialNumbers(0)
Sequence of Exponential numbers around 0
Bases: sage.combinat.sloane_functions.SloaneSequence
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
EXAMPLES:
sage: it = sloane.A000153.gen(0,1,2)
sage: [next(it) for i in range(5)]
[0, 1, 2, 7, 32]
EXAMPLES:
sage: sloane.A000153.list(8)
[0, 1, 2, 7, 32, 181, 1214, 9403]
Bases: sage.combinat.sloane_functions.ExtremesOfPermanentsSequence
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
EXAMPLES:
sage: from sage.combinat.sloane_functions import ExtremesOfPermanentsSequence2
sage: e = ExtremesOfPermanentsSequence2()
sage: it = e.gen(6,43,6)
sage: [next(it) for i in range(5)]
[6, 43, 307, 2542, 23799]
Bases: sage.combinat.sloane_functions.SloaneSequence
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
EXAMPLES:
sage: sloane.A001110.list(8)
[0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881]
Bases: sage.combinat.sloane_functions.SloaneSequence
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
EXAMPLES:
sage: sloane.A001906.list(10)
[0, 1, 3, 8, 21, 55, 144, 377, 987, 2584]
Bases: sage.structure.sage_object.SageObject
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:
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'>
Bases: sage.structure.sage_object.SageObject
Base class for a Sloane integer sequence.
EXAMPLES:
We create a dummy sequence:
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]
This functions calculates from Sloane’s sequences
A079908-A079928
INPUT:
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:
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: [next(it) for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
inhomogenous second-order linear recurrence generator with fixed
coefficients and
,
,
.
EXAMPLES:
sage: from sage.combinat.sloane_functions import recur_gen2b
sage: it = recur_gen2b(1,1,1,1, lambda n: 0)
sage: [next(it) for i in range(10)]
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
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: [next(it) for i in range(10)]
[1, 1, 1, 3, 5, 9, 17, 31, 57, 105]