A352419 Triangle read by rows T(n,k): number of three-in-a-rows in n-dimensional tic-tac-toe through a cell that is central in k dimensions (for k=0..n).

0, 1, 1, 3, 2, 4, 7, 4, 5, 13, 15, 8, 7, 14, 40, 31, 16, 11, 16, 41, 121, 63, 32, 19, 20, 43, 122, 364, 127, 64, 35, 28, 47, 124, 365, 1093, 255, 128, 67, 44, 55, 128, 367, 1094, 3280, 511, 256, 131, 76, 71, 136, 371, 1096, 3281, 9841, 1023, 512, 259, 140, 103, 152, 379, 1100, 3283, 9842, 29524

Offset: 0

Author

Ben Orlin, Mar 15 2022

Keywords

• nonn
• tabl

Comments

A tic-tac-toe board in n dimensions consists of 3^n cells. Each cell is central (between others) in k dimensions and extremal (not between others) in n-k dimensions. In standard n=2 tic-tac-toe, k=0 gives a corner, k=2 gives the center, and k=1 gives an edge.

A000225 gives the first term in each row: a(n) is the number of three-in-a-rows passing through corner cells in n-dimensional tic-tac-toe = 2^n - 1.

A003462 gives the final term in each row: a(n) is the number of three-in-a-rows passing through the center cell in n-dimensional tic-tac-toe = (3^n - 1)/2.

A007051 gives the penultimate term in each row: a(n) is the number of three-in-a-rows passing through a cell in n-dimensional tic-tac-toe that is central in n - 1 dimensions and extremal in 1 dimension = (3^(n-1))/2 + 1.

A170804 gives the minimum of each row: a(n) is the smallest number of three-in-a-rows passing through any cell in n-dimensional tic-tac-toe.

A094374 -1 gives the central values of even rows: a(n) - 1 is the number of three-in-a-rows passing through a cell in 2n-dimensional tic-tac-toe that is central in n dimensions and extremal in n dimensions = (2^n - 1) + (3^n - 1)/2.

Examples

			Table begins:
   0;
   1,  1;
   3,  2,  4;
   7,  4,  5, 13;
  15,  8,  7, 14, 40;
  31, 16, 11, 16, 41, 121;
  63, 32, 19, 20, 43, 122, 364;
		

Crossrefs

Cf. A000225, A003462, A007051, A094374, A170804.

Formula

T(n,k) = (3^k - 1)/2 + 2^(n-k) - 1.