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A243141 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.

%I A243141 #7 May 31 2014 15:52:04
%S A243141 1,1,1,2,4,3,1,3,10,19,22,7,1,4,22,75,170,204,115,18,1,5,41,218,816,
%T A243141 1891,2635,1909,628,58,3,7,72,542,2947,10846,26695,41770,39218,19905,
%U A243141 4776,437,13,8,116,1178,8765,46068,171700,444117,776276,876012,601078,229941
%N A243141 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.
%C A243141 The triangle T(n, k) is irregularly shaped: 1 <= k <= A240114(n). First row corresponds to n = 1.
%C A243141 The maximal number of points that can be placed on a triangular grid of side n so that no three of them form an equilateral triangle is given by A240114(n).
%H A243141 Heinrich Ludwig, <a href="/A243141/b243141.txt">Table of n, a(n) for n = 1..129</a>
%e A243141 The triangle begins:
%e A243141   1;
%e A243141   1,  1;
%e A243141   2,  4,   3,    1;
%e A243141   3, 10,  19,   22,     7,     1;
%e A243141   4, 22,  75,  170,   204,   115,    18,     1;
%e A243141   5, 41, 218,  816,  1891,  2635,  1909,   628,    58,    3;
%e A243141   7, 72, 542, 2947, 10846, 26695, 41770, 39218, 19905, 4776, 437, 13;
%e A243141   ...
%e A243141 There is exactly T(5, 8) = 1 way to place 8 points (x) on a triangular grid of side 5 according to the definition of the sequence:
%e A243141            .
%e A243141           x x
%e A243141          x . x
%e A243141         x . . x
%e A243141        x . . . x
%Y A243141 Cf. A240114, A240439, A001399 (column 1), A227327 (column 2), A243142 (column 3), A243143 (column 4), A243144 (column 5).
%K A243141 nonn,tabf
%O A243141 1,4
%A A243141 _Heinrich Ludwig_, May 30 2014