cp's OEIS Frontend

Rows are numbers of dominoes with k spots where each half-domino has zero to n spots (in standard domino set: n=6, there are 28 dominoes and row is 1,1,2,2,3,3,4,3,3,2,2,1,1). - Henry Bottomley, Aug 23 2000

The Gaussian polynomial (or Gaussian binomial) [n,2]q is an example of a q-binomial coefficient (see the link) and may be defined for n >= 2 by [n,2]_q = ([n]_q * [n-1]_q)/([1]_q * [2]_q), where [n]_q := q^n - 1. The first few values are: [2,2]_q = 1; [3,2]_q = 1 + q + q^2; [4,2]_q = 1 + q + 2q^2 + q^3 + q^4. - _Peter Bala, Sep 23 2007

These numbers appear in the solution of Cayley's counting problem on covariants as N(p,2,w) = [x^p,q^w] Phi(q,x) with the o.g.f. Phi(q,x) = 1/((1-x)(1-qx)(1-q^2x)) given by Peter Bala in the formula section. See the Hawkins reference, p. 264, were also references are given. - Wolfdieter Lang, Nov 30 2012

The entry a(p,w), p >= 0, w = 0,1,...,2*p, of this irregular triangle is the number of nonnegative solutions of m_0 + m_1 + m_2 = p and 1*m_1 + 2*m_2 = w. See the Hawkins reference p. 264, (4.8). N(p,2,w) there is a(p,w). See also the Cayley reference p. 110, 35. with m = 2, Theta = p and q = w. - Wolfdieter Lang, Dec 01 2012

From Gus Wiseman, Sep 20 2023: (Start)

Also the number of unordered pairs of distinct positive integers up to n with sum k. For example, row n = 9 counts the following pairs:

12 13 14 15 16 17 18 19 29 39 49 59 69 79 89

23 24 25 26 27 28 38 48 58 68 78

34 35 36 37 47 57 67

45 46 56

Allowing repeated parts (x,x) gives A004737.

For strict partitions instead of just pairs we have A053632.

(End)

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

a(n) is 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, g5 and h6; Black Ke8, Nh8, pawns b3, c7, d3, f7, g6 and h7. (After Noam D. Elkies, see link; diagram 5).

Counts all paths of length n, n >= 0, starting at the third node on the path graph P_5, see the Maple program. (End)

From Alec Jones, Feb 25 2016: (Start)

The a(n) are the n-th terms in a "Fibonacci snake" drawn on a rectilinear grid. The n-th term is computed as the sum of the previous terms in cells adjacent to the n-th cell (diagonals included). (This sequence excludes the first term of the snake.)

For example:

1 ... 1 1 ... 1 4 1 4 6 ... 1 4 6 1 4 6 ... and so on.

1 ... 1 2 1 2 ... 1 2 1 2 12 ... 1 2 12 18 (End)

From Gus Wiseman, Oct 06 2023: (Start)

Also the number of subsets of {1..n} containing no two distinct elements summing to n. The a(0) = 1 through a(4) = 12 subsets are:

{} {} {} {} {}

{1} {1} {1} {1}

{2} {2} {2}

{1,2} {3} {3}

{1,3} {4}

{2,3} {1,2}

{1,4}

{2,3}

{2,4}

{3,4}

{1,2,4}

{2,3,4}

For n+1 instead of n we have A038754, complement A167762.

Including twins gives A117855, complement A366131.

The complement is counted by A365544.

For all subsets (not just pairs) we have A365377, complement A365376. (End)

Rows are palindromic.

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

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)

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]

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

Also the number of integer partitions of n with no two possibly equal parts summing to n.