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Triangle T(n, k) = Number of ways to choose k points from an n X n X n triangular grid so that no three of them form a 2 X 2 X 2 subtriangle. Triangle T read by rows.
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%I #17 Feb 01 2015 05:15:01

%S 1,1,1,3,3,1,6,15,16,6,1,10,45,111,156,120,42,2,1,15,105,439,1191,

%T 2154,2583,1977,885,189,9,1,21,210,1305,5565,17052,38337,63576,77208,

%U 67285,40512,15750,3480,333,9,1,28,378,3240,19620,88590,307362,833228,1779219

%N Triangle T(n, k) = Number of ways to choose k points from an n X n X n triangular grid so that no three of them form a 2 X 2 X 2 subtriangle. Triangle T read by rows.

%C n starts from 1. The maximal number of points that can be chosen from a grid of side n, so that no three of them are forming a subtriangle of side 2, is A007980(n - 1). So k ranges from 0 to A007980(n - 1).

%C Column #2 (k = 1) is A000217.

%C Column #3 (k = 2) is A050534.

%C Column #4 (k = 3) is A234250.

%H Heinrich Ludwig, <a href="/A234251/b234251.txt">Table of n, a(n) for n = 1..133</a>

%e Triangle begins

%e 1, 1;

%e 1, 3, 3;

%e 1, 6, 15, 16, 6;

%e 1, 10, 45, 111, 156, 120, 42, 2;

%e 1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9;

%e ...

%e There are no more than T(4, 7) = 2 ways to choose 7 points (X) from a 4 X 4 X 4 grid so that no 3 of them form a 2 X 2 X 2 subtriangle:

%e X X

%e X . . X

%e . X X X X .

%e X X . X X . X X

%Y Cf. A000217, A050534, A234250.

%K nonn,tabf

%O 1,4

%A _Heinrich Ludwig_, Feb 06 2014