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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 with sides parallel to the grid. Triangle read by rows.
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%I #9 Jun 13 2014 12:04:08

%S 1,1,1,3,3,1,6,15,15,3,1,10,45,107,128,63,10,1,15,105,428,1062,1566,

%T 1276,507,69,1,21,210,1282,5160,13971,25191,29235,20508,7747,1251,42,

%U 1,1,28,378,3198,18591,77124,231090,498097,759117,792942,540361,222597,49053

%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 with sides parallel to the grid. Triangle read by rows.

%C The triangle T(n, k) is irregularly shaped: 0 <= k <= A227308(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 with sides parallel to the grid is given by A227308(n).

%H Heinrich Ludwig, <a href="/A243211/b243211.txt">Table of n, a(n) for n = 1..165</a>

%e The triangle begins:

%e 1, 1;

%e 1, 3, 3;

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

%e 1, 10, 45, 107, 128, 63, 10,

%e 1, 15, 105, 428, 1062, 1566, 1276, 507, 69,

%e 1, 21, 210, 1282, 5160, 13971, 25191, 29235, 20508, 7747, 1251, 42, 1;

%e ...

%e There is T(6, 12) = 1 way to place 12 points (x) on the grid obeying the rule in the definition of the sequence:

%e .

%e x x

%e x . x

%e x . . x

%e x . . . x

%e . x x x x .

%Y Cf. A227308, A243207, A084546, A234251, A239567, A240439, A194136, A000217 (column 2), A050534 (column 3), A243212 (column 4), A243213 (column 5), A243214 (column 6).

%K nonn,tabf

%O 1,4

%A _Heinrich Ludwig_, Jun 09 2014