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)

Row n has length 2n+1.

Rules #50, #58, #114, #122, #178, #179, #186, #242, #250 all give rise to this sequence.

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446, A080883. - Jeremy Gardiner, Dec 30 2006

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]

This sequence contains every integer infinitely often, hence all integer sequences are subsequences.

This is a fractal sequence.

Indeed, if all terms (a(n),a(n+1)) such that a(n+1) does NOT equal a(n)+1 (<=> a(n+1) < a(n)) are deleted, the same sequence is recovered again. See A253580 for an "opposite" yet similar "fractal tree" construction. - M. F. Hasler, Jan 04 2015

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

Sum of n-th row is n^3. The n-th row contains 3n-2 entries. Largest coefficients in rows are listed in A077043. The 255th row describes the distribution of color lattice points in the 765 r+g+b=k planes of the 24-bit RGB-cube with 256^3 points.

Also, the number of cubes of dimension 1 X 1 X 1 needed to build a cube by layers perpendicular to the main diagonal. Each layer is made up of regular triangular numbers T near the summits and truncated T's in the middle. E.g., cube 3^3 is made of layers 1, 3, 6, 7, 6, 3, 1, using T1, T2, T3 and a regularly truncated T4, 7 instead of 10. - M. Dauchez (mdzzdm(AT)yahoo.fr), Aug 31 2005

The n-th row is the third row of the (n+1)-nomial triangle. For example, row 1 (1,3,3,1) is the third row in the binomial triangle; row 5 is the third row of the 6-nomial triangle. - Bob Selcoe, Feb 18 2014

It appears that T(n,k) gives the number of possible ways of randomly selecting k cards from n-1 sets, each with three different playing cards. - Juan Pablo Herrera P., Nov 04 2016

The triangle A003983 with individual entries squared and each 2nd row skipped.

Analogous to A004737. - Peter Bala, Sep 25 2007

T(n,k) = min(n,k)^2. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

a(A000326(n)) = a(A143689(n-1)) = 1;

for all n: a(k) = n, A104249(n-1) <= k <= A005448(n).

Equivalently, T(k,n,h) is the number of ordered sets of n nonnegative integers < k with the sum equal to h.

From Juan Pablo Herrera P., Nov 21 2016: (Start)

T(k,n,h) is the number of possible ways of randomly selecting h cards from k-1 sets, each with n different playing cards. It is also the number of lattice paths from (0,0) to (n,h) using steps (1,0), (1,1), (1,2), ..., (1,k-1).

Shallow diagonal sums of each triangle with fixed k give the k-bonacci numbers. (End)

T(k,n,h) is the number of n-dimensional grid points of a k X k X ... X k grid, which are lying in the (n-1)-dimensional hyperplane which is at an L1 distance of h from one of the grid's corners, and normal to the corresponding main diagonal of the grid. - Eitan Y. Levine, Apr 23 2023