cp's OEIS Frontend

a(n+2) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = A000110(k) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003

Number of complete rhyming schemes.

Whereas the Bell number B(n) (A000110(n)) is the number of terms in the polynomial that expresses the n-th moment of a probability distribution as a function of the first n cumulants, these numbers give the number of terms in the corresponding expansion of the central moment as a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), Jan 26 2005

a(n) is the number of permutations on [n] for which the left-to-right maxima coincide with the descents (entries followed by a smaller number). For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan, Jul 20 2005

From Gus Wiseman, Feb 10 2019: (Start)

Also the number of stable partitions of an n-cycle, where a stable partition of a graph is a set partition of the vertex set such that no edge has both ends in the same block. A bijective proof is given in David Callan's article. For example, the a(5) = 11 stable partitions are:

{{1},{2},{3},{4},{5}}

{{1},{2},{3,5},{4}}

{{1},{2,4},{3},{5}}

{{1},{2,5},{3},{4}}

{{1,3},{2},{4},{5}}

{{1,4},{2},{3},{5}}

{{1},{2,4},{3,5}}

{{1,3},{2,4},{5}}

{{1,3},{2,5},{4}}

{{1,4},{2},{3,5}}

{{1,4},{2,5},{3}}

(End)

Also number of partitions of {1, 2, ..., n-1} with singletons. E.g., a(4) = 4: {1|2|3, 12|3, 13|2, 1|23}. Also number of cyclical adjacencies partitions of {1, 2, ..., n-1}. E.g., a(4) = 4: {12|3, 13|2, 1|23, 123}. The two partitions can be mapped by a Kreweras bijection. - Yuchun Ji, Feb 22 2021

Also the k-th central moment of a Poisson random variable with mean 1. a(n) = E[(X-1)^n, X~Poisson(1)]. - Thomas Dybdahl Ahle, Dec 14 2022

For prime p, p divides a(p-1). - T. D. Noe, Apr 11 2009 [This result follows immediately from the fact that A032190(n) = (1/n)*Sum_{d|n} a(d-1)*phi(n/d). - Petros Hadjicostas, Sep 11 2017]

Generalization. If a(0,x)=0, a(1,x)=2 and, for n>=2, a(n,x)=a(n-1,x)+x*a(n-2,x)+1, then we obtain a sequence of polynomials Q_n(x)=a(n,x) of degree floor((n-1)/2), such that p is prime iff all coefficients of Q_(p-1)(x) are multiple of p (sf. A174625). Thus a(n) is the sum of coefficients of Q_(n-1)(x). - Vladimir Shevelev, Apr 23 2010

Odd composite numbers n such that n divides a(n-1) are in A005845. - Zak Seidov, May 04 2010; comment edited by N. J. A. Sloane, Aug 10 2010

a(n) is the number of ways to modify a circular arrangement of n objects by swapping one or more adjacent pairs. E.g., for 1234, new arrangements are 2134, 2143, 1324, 4321, 1243, 4231 (taking 4 and 1 to be adjacent) and a(4) = 6. - Toby Gottfried, Aug 21 2011

For n>2, a(n) equals the number of Markov equivalence classes with skeleton the cycle on n+1 nodes. See Theorem 2.1 in the article by A. Radhakrishnan et al. below. - Liam Solus, Aug 23 2018

From Gus Wiseman, Feb 12 2019: (Start)

For n > 0, also the number of nonempty subsets of {1, ..., n + 1} containing no two cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(5) = 17 stable subsets are:

{1}, {2}, {3}, {4}, {5}, {6},

{1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},

{1,3,5}, {2,4,6}.

(End)

Also the rank of the n-Lucas cube graph. - Eric W. Weisstein, Aug 01 2023

From Michael Somos, Mar 04 2004: (Start)

Stirling transform of A005359(n)=[0,2,0,24,0,720,...] is a(n)=[0,2,6,38,270,...].

Stirling transform of -(-1)^n*A052657(n-1)=[0,0,2,-6,48,-240,...] is a(n-1)=[0,0,2,6,38,270,...].

Stirling transform of -(-1)^n*A052558(n-1)=[1,-1,4,-12,72,-360,...] is a(n-1)=[1,0,2,6,38,270,...].

Stirling transform of 2*A052591(n)=[2,4,24,96,...] is a(n+1)=[2,6,38,270,...].

(End)

Also the central moments of a Geometric(1/2) random variable (for example the number of coin tosses until the first head). - Svante Janson, Dec 10 2012

Also the number of ordered set partitions of {1..n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). - Gus Wiseman, Feb 13 2019

Also the number of ordered set partitions of {1..n} with an even number of blocks. - Geoffrey Critzer, Jul 04 2020

a(n)-1 counts ternary numbers with no 0 digit (A007931) and at least one 2 digit, where the total of ternary digits is <= n. E.g., a(4)-1 = 7: 2 12 21 22 112 121 211. - Frank Ellermann, Dec 02 2001

A107909(a(n-1)) = A000079(n-1) = 2^(n-1). - Reinhard Zumkeller, May 28 2005

a(n) is the permanent of the n X n 0-1 matrix whose (i,j) entry is 1 iff i=1 or j=n or |i-j|<=1. For example, a(5)=15 is per([[1, 1, 1, 1, 1], [1, 1, 1, 0, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1], [0, 0, 0, 1, 1]]). - David Callan, Jun 07 2006

Conjecture. Let S(1)={1} and, for n>1, let S(n) be the smallest set containing x+1 and 2x+1 for each element x in S(n-1). Then a(n) is the sum of the elements in S(n). (See A122554 for a sequence defined in this way.) - John W. Layman, Nov 21 2007

a(n+1) indexes the corner blocks on the Fibonacci spiral built from blocks of unit area (using F(1) and F(2) as the sides of the first block). - Paul Barry, Mar 06 2008

The number of length n binary words with fewer than 2 0-digits between any pair of consecutive 1-digits. - Jeffrey Liese, Dec 23 2010

If b(n) = a(n+1) then b(0) = 1 and 2*b(n) >= b(n+1) for all n > 1 which is sufficient for b(n) to be a complete sequence. - Frank M Jackson, Mar 17 2013

From Gus Wiseman, Feb 10 2019: (Start)

Also the number of non-singleton subsets of {1, ..., n + 1} with no successive elements. For example, the a(5) = 15 subsets are:

{},

{1,3}, {1,4}, {1,5}, {1,6}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},

{1,3,5}, {1,3,6}, {1,4,6}, {2,4,6}.

Also the number of binary sequences with all zeros or at least 2 ones and no adjacent ones. For example, the a(1) = 1 through a(4) = 8 sequences are:

(00) (000) (0000) (00000)

(101) (0101) (00101)

(1001) (01001)

(1010) (01010)

(10001)

(10010)

(10100)

(10101)

(End)

a(n) = floor(lim_{k->oo} Fibonacci(k)/Fibonacci(k-n)). - Jon Perry, Jun 10 2003

For n > 1, a(n) is the maximum element in the continued fraction for A000045(n)*phi. - Benoit Cloitre, Jun 19 2005

a(n) is also the curvature (rounded down) of the circle inscribed in the n-th kite arranged in a spiral, starting with a unit circle, as shown in the illustration in the links section. - Kival Ngaokrajang, Aug 29 2013

a(n) is the n-th Lucas number (A000032) if n is odd, and a(n) is the n-th Lucas number minus 1 if n is even. (Mario Catalani's formula below expresses this fact.) This is related to the fact that the powers of phi approach the values of the Lucas numbers, the odd powers from above and the even powers from below. - Geoffrey Caveney, Apr 18 2014

a(n) is the sum of the last summands over all Arndt compositions of n (see the Checa link). - Daniel Checa, Dec 25 2023

a(n) is the number of (saturated or unsaturated) substituted N-heterocycles in chemistry (N = nitrogen). That means the number of matchings in a cycle graph when the two maximum matchings in every cycle with an even number of vertices are indistinguishable (because the corresponding resonance structures in the molecule are equivalent). - Stefan Schuster, Mar 20 2025

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).

Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009

Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014

Also the number of set partitions of {1, ..., n} with k cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). Unlike A250104, we count {{1}} as having 1 cyclical adjacency. - Gus Wiseman, Feb 13 2019

Even-numbered terms a(2k) are A002872: 2,7,31,164,999 ("Sorting numbers"); odd-numbered terms are its binomial transform, A080337. The symmetrical set partitions of {-n,...,-1,0,1,...,n} can be classified by the partition containing 0. Thus we get the sum over k of {n choose k} times the number of symmetrical set partitions of 2n-2k elements. - Don Knuth, Nov 23 2003

Number of partitions of n numbers that are symmetrical and cannot be nested (i.e., include a pattern of the form abab). - Douglas Boffey, May 21 2015

Number of achiral color patterns in a row or loop of length n. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 23 2018

Also the number of self-complementary set partitions of {1, ..., n}. The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. - Gus Wiseman, Feb 13 2019

From Gus Wiseman, Feb 12 2019: (Start)

Also the number of ways to split an (n + 1)-cycle into nonempty connected subgraphs with no singletons. For example, the a(1) = 1 through a(5) = 12 partitions are:

{{12}} {{123}} {{1234}} {{12345}} {{123456}}

{{12}{34}} {{12}{345}} {{12}{3456}}

{{14}{23}} {{123}{45}} {{123}{456}}

{{125}{34}} {{1234}{56}}

{{145}{23}} {{1236}{45}}

{{15}{234}} {{1256}{34}}

{{126}{345}}

{{1456}{23}}

{{156}{234}}

{{16}{2345}}

{{12}{34}{56}}

{{16}{23}{45}}

Also the number of non-singleton subsets of {1, ..., (n + 1)} with no cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(1) = 1 through a(5) = 12 subsets are:

{} {} {} {} {}

{1,3} {1,3} {1,3}

{2,4} {1,4} {1,4}

{2,4} {1,5}

{2,5} {2,4}

{3,5} {2,5}

{2,6}

{3,5}

{3,6}

{4,6}

{1,3,5}

{2,4,6}

(End)

These set partitions are fixed points under Callan's bijection phi on set partitions.