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Triangle T(n, k) = Numbers of 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.
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%I #9 Apr 06 2014 03:27:06

%S 1,1,1,3,3,1,6,15,15,3,1,10,45,105,114,39,3,1,15,105,420,969,1194,654,

%T 102,3,1,21,210,1260,4773,11259,15615,11412,3663,342,15,1,28,378,3150,

%U 17415,64776,159528,250233,234609,119259,28395,2613,69,1,36,630,6930

%N Triangle T(n, k) = Numbers of 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: 0 <= 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="/A240439/b240439.txt">Table of n, a(n) for n = 1..138</a>

%e The triangle begins:

%e 1, 1;

%e 1, 3, 3;

%e 1, 6, 15, 15, 3;

%e 1, 10, 45, 105, 114, 39, 3;

%e 1, 15, 105, 420, 969, 1194, 654, 102, 3;

%e 1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15;

%e There are T(5, 8) = 3 ways to place 8 points (x) on a triangular grid of side 5 under the conditions mentioned above:

%e . x x

%e x x x . . x

%e x . x x . . . . x

%e x . . x x . . . . . . x

%e x . . . x . x x x x x x x x .

%Y Cf. A240114, A240443, A084546,

%Y column 2 is A000217,

%Y column 3 is A050534,

%Y column 4 is A240440,

%Y column 5 is A240441,

%Y column 6 is A240442.

%K nonn,tabf

%O 1,4

%A _Heinrich Ludwig_, Apr 05 2014