cp's OEIS Frontend

Poly-Bernoulli numbers B_n^(k) with k=-2.

Binomial transform of A007051, if both sequences start at 0. Binomial transform of A000225(n+1). - Paul Barry, Mar 24 2003

Euler expands (1-z)/(1-5z+6z^2) and finds the general term. Section 226 of the Introductio indicates that he could have written down the recursion relation: a(n) = 5 a(n-1)-6 a(n-2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006

Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xRy if x is a subset of y or y is a subset of x. Then a(n) = |R|. - Ross La Haye, Dec 22 2006

With regard to the comment by Ross La Haye: For proper subsets see A056182. - For nonempty subsets see A091344. - For nonempty proper subsets see a(n+1) in A260217. - Manfred Boergens, Aug 02 2023

If x, y are two n-bit binary strings then a(n) gives the number of pairs (x,y) such that XOR(x, y) = ABS(x - y). - Ramasamy Chandramouli, Feb 15 2009

Equals row sums of the triangular version of A038573. - Gary W. Adamson, Jun 04 2009

Inverse binomial transform of A085350. - Paul Curtz, Nov 14 2009

Related to the number of even a's in a nontrivial cycle (should one exist) in the 3x+1 Problem, where a <= floor(log_2(2*(3^n) - 2^n)). The value n correlates to the number of odds in such a nontrivial cycle. See page 1288 of Crandall's paper. Also, this relation gives another proof that the number of odds divided by the number of evens in a nontrivial cycle is bounded by log 2 / log 3 (this observation does not resolve the finite cycles conjecture as the value could be arbitrarily close to this bound). However, the same argument gives that log 2 / log 3 is less than or equal to the number of odds divided by the number of evens in a divergent sequence (should one exist), as log 2 / log 3 is the limit value for a cycle of an arbitrarily large length, where the length is given by the value n. - Jeffrey R. Goodwin, Aug 04 2011

Row sums of Riordan triangle A106516. - Wolfdieter Lang, Jan 09 2015

Number of restricted barred preferential arrangements having 3 bars in which the sections are all restricted sections such that (for fixed sections i and j) section i or section j is empty. - Sithembele Nkonkobe, Oct 12 2015

This is also row 2 of A281891: for n >= 1, when consecutive positive integers are written as a product of primes in nondecreasing order, a factor of 2 or 3 occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 18 2017

Also row sums of A124929. - Omar E. Pol, Jun 15 2017

This is the sum of A318921(n) for n in the range 2^(k+1) to 2^(k+2)-1. See A318921 for proof. - N. J. A. Sloane, Sep 25 2018

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{2,n}. - Vincent Pilaud, Sep 15 2020

a(n-1) is also the number of n-digit numbers whose largest decimal digit is 2. - Stefano Spezia, Nov 15 2023

Columns 0,1,2,3 respectively correspond to relations R_3, R_4, R_0, R_1 defined in La Haye paper listed below.

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner or side square (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white chess queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.

On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the central square (we assume here that a red queen might behave like a white queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner and side squares the 512 red queens lead to 17 red queen sequences, see the cross-references for the complete set.

The sequence above corresponds to 8 red queen vectors, i.e., A[5] vectors, with decimal values 239, 367, 431, 463, 487, 491, 493 and 494. The central square leads for these vectors to A152240.

This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 5*x - k*x^2). The members of this family that are red queen sequences are A180030 (k=8), A180032 (k=7; this sequence), A000400 (k=6), A180033 (k=5), A126501 (k=4), A180035 (k=3), A180037 (k=2) A015449 (k=1) and A003948 (k=0). Other members of this family are A030221 (k=-1), A109114 (k=-3), A020989 (k=-4), A166060 (k=-6).

Inverse binomial transform of A054413.

All rows except the zeroth are divisible by 3. Is there a closed-form formula for these numbers, like for binomial coefficients?

Let b=3 and T(n,k) = A(n-k,k) be the associated reading of the symmetric array A by antidiagonals, then A(n,k) = sum_{r=1..n} b^r*A178300(n-r,k) + sum_{c=1..k} b^c*A178300(k-c,n). Similarly with b=4 and b=5 for A227074 and A227076. - R. J. Mathar, Aug 10 2013

b(1)=1, b(n)=2*b(n/2) if n is even, or b(n)=b(n-1)+1 if n is odd produces the sequence of natural numbers.

Seen as a triangle read by rows: T(1,1) = 1; T(n+1,2*k-1) = T(n,k)+1 and T(n+1,2*k) = 2*T(n,k)+2, 1 <= k <= 2^n. - Reinhard Zumkeller, May 13 2015

The a(n) represent the number of n-move paths of a chess queen starting or ending in the central square (m = 5) on a 3 X 3 chessboard. The other squares lead to A180030.

To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the queen's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.

Closely related with this sequence are the red queen sequences, see A180028 and A180032.

This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 5*x - (k+5)*x^2). The members of this family that are red queen sequences are A180031 (k=3; this sequence), A152240 (k=2), A000400 (k=1), A057088 (k=0), A122690 (k=-1), A180036 (k=-2), A180038 (k=-3), A015449 (k=-4) and A000007 (k=-5). Other members of this family are A030221 (k= -6), 3*A109114 (k=-8), 4*A020989 (k=-9), 6*A166060 (k=-11).

a(n) <= A166060(n).