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A234247 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.

%I A234247 #24 Feb 23 2014 10:03:19
%S A234247 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 A234247 41,231,955,2880,6459,10660,12948,11274,6802,2645,595,57,2,7,72,566,
%U A234247 3335,14883,51470,139224,297048,500147,661796,681101,536322,314753,132490
%N A234247 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 A234247 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 A234247 Column #1 (k = 1) is A001399.
%C A234247 Column #2 (k = 2) is A227327.
%C A234247 Without the restriction "non-equivalent (mod D_3)" numbers are given by A234251.
%H A234247 Heinrich Ludwig, <a href="/A234247/b234247.txt">Table of n, a(n) for n = 1..123</a>
%e A234247 Triangle begins
%e A234247 1;
%e A234247 1,  1;
%e A234247 2,  4,   4,   2;
%e A234247 3, 10,  22,  31,   22,   10,     1;
%e A234247 4, 22,  82, 212,  374,  450,   342,   156,    36,    2;
%e A234247 5, 41, 231, 955, 2880, 6459, 10660, 12948, 11274, 6802, 2645, 595, 57, 2;
%e A234247 ...
%e A234247 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 A234247        .                X
%e A234247       X X              . X
%e A234247      X . X            X . X
%e A234247     . X X .          . X X .
%e A234247    X X . X X        X X . X X
%Y A234247 Cf. A234251, A007980, A001399, A227327.
%K A234247 nonn,tabf
%O A234247 1,4
%A A234247 _Heinrich Ludwig_, Feb 11 2014