cp's OEIS Frontend

Inverse binomial transform yields two zeros followed by A077917 (a signed variant of A127864).

a(n) mod 10 is zero followed by a sequence with period length 8: 0, 1, 2, 7, 4, 7, 4, 5 (repeat).

a(n) is the number of length n+1 binary words with some prefix w such that w contains three more 1's than 0's and no prefix of w contains three more 0's than 1's. - Geoffrey Critzer, Dec 13 2013

From Gus Wiseman, Oct 06 2023: (Start)

Also the number of subsets of {1..n} with two distinct elements summing to n + 1. For example, the a(2) = 1 through a(5) = 14 subsets are:

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

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

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

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

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

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

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

{2,4,5}

{1,2,3,4}

{1,2,3,5}

{1,2,4,5}

{1,3,4,5}

{2,3,4,5}

{1,2,3,4,5}

The complement is counted by A038754.

Allowing twins gives A167936, complement A108411.

For n instead of n + 1 we have A365544, complement A068911.

The version for all subsets (not just pairs) is A366130.

(End)

Case for base 2 see A000225: 2^n - 1.

If the sequence b(n) represents the number of paths of length n, n >= 1, starting at node 1 and ending at nodes 1, 2, 3 and 4 on the path graph P_5 then a(n-1) = b(n) - 1. - Johannes W. Meijer, May 29 2010

See A225367 for the sequence that counts all base 3 palindromes, including 0 (and thus also the number of n-digit terms in A006072). -- A nonzero palindrome of length L=2k-1 or of length L=2k is determined by the first k digits, which then determine the last k digits by symmetry. Since the first digit cannot be 0, there are 2*3^(k-1) possibilities. - M. F. Hasler, May 05 2013

From Gus Wiseman, Oct 18 2023: (Start)

Also the number of subsets of {1..n} with n not the sum of two subset elements (possibly the same). For example, the a(0) = 1 through a(4) = 6 subsets are:

{} {} {} {} {}

{1} {2} {1} {1}

{2} {3}

{3} {4}

{1,3} {1,4}

{2,3} {3,4}

For subsets with no subset summing to n we have A365377.

Requiring pairs to be distinct gives A068911, complement A365544.

The complement is counted by A366131.

(End) [Edited by Peter Munn, Nov 22 2023]

From Johannes W. Meijer, May 29 2010: (Start)

The absolute values of the a(n) represent the number of ways White can force checkmate in exactly (n+1) moves, n>=0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6 and h6; Black Ke8, pawns b3, c7, d3, f7 and h7. (After Noam D. Elkies, see link; diagram 5).

The absolute values of the a(n) represent all paths of length n starting at the third (or fourth) node on the path graph P_6, see the Maple program.

(End)

For n>=1, abs(a(n-1)) is the number of compositions where there is no rise between every second pair of parts, starting with the second and third part; see example. Also, abs(a(n-1)) is the number of compositions of n where there is no fall between every second pair of parts, starting with the second and third part; see example. [Joerg Arndt, May 21 2013]

Also the number of strict integer partitions of n containing two possibly equal elements summing to n.