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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).

%I A352419 #18 Apr 23 2022 14:41:38
%S A352419 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,
%T A352419 122,364,127,64,35,28,47,124,365,1093,255,128,67,44,55,128,367,1094,
%U A352419 3280,511,256,131,76,71,136,371,1096,3281,9841,1023,512,259,140,103,152,379,1100,3283,9842,29524
%N 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).
%C A352419 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.
%C A352419 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.
%C A352419 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.
%C A352419 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.
%C A352419 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.
%C A352419 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.
%F A352419 T(n,k) = (3^k - 1)/2 + 2^(n-k) - 1.
%e A352419 Table begins:
%e A352419    0;
%e A352419    1,  1;
%e A352419    3,  2,  4;
%e A352419    7,  4,  5, 13;
%e A352419   15,  8,  7, 14, 40;
%e A352419   31, 16, 11, 16, 41, 121;
%e A352419   63, 32, 19, 20, 43, 122, 364;
%Y A352419 Cf. A000225, A003462, A007051, A094374, A170804.
%K A352419 nonn,tabl
%O A352419 0,4
%A A352419 _Ben Orlin_, Mar 15 2022