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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.
5

%I #7 May 31 2014 15:52:04

%S 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 1891,2635,1909,628,58,3,7,72,542,2947,10846,26695,41770,39218,19905,

%U 4776,437,13,8,116,1178,8765,46068,171700,444117,776276,876012,601078,229941

%N 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 The triangle T(n, k) is irregularly shaped: 1 <= k <= A240114(n). First row corresponds to n = 1.

%C 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 Heinrich Ludwig, <a href="/A243141/b243141.txt">Table of n, a(n) for n = 1..129</a>

%e The triangle begins:

%e 1;

%e 1, 1;

%e 2, 4, 3, 1;

%e 3, 10, 19, 22, 7, 1;

%e 4, 22, 75, 170, 204, 115, 18, 1;

%e 5, 41, 218, 816, 1891, 2635, 1909, 628, 58, 3;

%e 7, 72, 542, 2947, 10846, 26695, 41770, 39218, 19905, 4776, 437, 13;

%e ...

%e 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 .

%e x x

%e x . x

%e x . . x

%e x . . . x

%Y Cf. A240114, A240439, A001399 (column 1), A227327 (column 2), A243142 (column 3), A243143 (column 4), A243144 (column 5).

%K nonn,tabf

%O 1,4

%A _Heinrich Ludwig_, May 30 2014