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Triangle T(n,k) read by rows: Number of non-equivalent ways (mod D_3) to choose k points from an nXnXn triangular grid so that no three of them form a 2X2X2 subtriangle.
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%I #24 Feb 23 2014 10:03:19

%S 1,1,1,2,4,4,2,3,10,22,31,22,10,1,4,22,82,212,374,450,342,156,36,2,5,

%T 41,231,955,2880,6459,10660,12948,11274,6802,2645,595,57,2,7,72,566,

%U 3335,14883,51470,139224,297048,500147,661796,681101,536322,314753,132490

%N Triangle T(n,k) read by rows: Number of non-equivalent ways (mod D_3) to choose k points from an nXnXn triangular grid so that no three of them form a 2X2X2 subtriangle.

%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 1 to A007980(n - 1).

%C Column #1 (k = 1) is A001399.

%C Column #2 (k = 2) is A227327.

%C Without the restriction "non-equivalent (mod D_3)" numbers are given by A234251.

%H Heinrich Ludwig, <a href="/A234247/b234247.txt">Table of n, a(n) for n = 1..123</a>

%e Triangle begins

%e 1;

%e 1, 1;

%e 2, 4, 4, 2;

%e 3, 10, 22, 31, 22, 10, 1;

%e 4, 22, 82, 212, 374, 450, 342, 156, 36, 2;

%e 5, 41, 231, 955, 2880, 6459, 10660, 12948, 11274, 6802, 2645, 595, 57, 2;

%e ...

%e There are exactly T(5, 10) = 2 non-equivalent ways to choose 10 points (X) from a triangular grid of side 5 avoiding that any three of them form a subtriangle of side 2.

%e . X

%e X X . X

%e X . X X . X

%e . X X . . X X .

%e X X . X X X X . X X

%Y Cf. A234251, A007980, A001399, A227327.

%K nonn,tabf

%O 1,4

%A _Heinrich Ludwig_, Feb 11 2014