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

1,4

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.

Links

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

Example

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

Crossrefs

Cf. A000217, A050534, A234250.

Sequence in context: A199034 A138464 A117279 * A049323 A322148 A084144

Adjacent sequences: A234248 A234249 A234250 * A234252 A234253 A234254

Keyword

nonn,tabf

Author

Heinrich Ludwig, Feb 06 2014

Status

approved