A234251 Triangle T(n, k) = Number of ways to choose k points from an n X n X n triangular grid so that no three of them form a 2 X 2 X 2 subtriangle. Triangle T read by rows.

1, 1, 1, 3, 3, 1, 6, 15, 16, 6, 1, 10, 45, 111, 156, 120, 42, 2, 1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9, 1, 21, 210, 1305, 5565, 17052, 38337, 63576, 77208, 67285, 40512, 15750, 3480, 333, 9, 1, 28, 378, 3240, 19620, 88590, 307362, 833228, 1779219

Offset: 1

Author

Heinrich Ludwig, Feb 06 2014

Keywords

• nonn
• tabf

Comments

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 0 to A007980(n - 1).

Column #2 (k = 1) is A000217.

Column #3 (k = 2) is A050534.

Column #4 (k = 3) is A234250.

Examples

			Triangle begins
  1,  1;
  1,  3,   3;
  1,  6,  15,  16,    6;
  1, 10,  45, 111,  156,  120,   42,    2;
  1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9;
  ...
There are no more than T(4, 7) = 2 ways to choose 7 points (X) from a 4 X 4 X 4 grid so that no 3 of them form a 2 X 2 X 2 subtriangle:
        X              X
       X .            . X
      . X X          X X .
     X X . X        X . X X
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..133

Crossrefs

Cf. A000217, A050534, A234250.