OFFSET
0,4
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.
FORMULA
T(n,k) = (3^k - 1)/2 + 2^(n-k) - 1.
EXAMPLE
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
KEYWORD
nonn,tabl
AUTHOR
Ben Orlin, Mar 15 2022
STATUS
approved