A234247 Triangle T(n,k) read by rows: Number of non-equivalent ways (mod D_3) to choose k points from an nXnXn triangular grid so that no three of them form a 2X2X2 subtriangle.

1, 1, 1, 2, 4, 4, 2, 3, 10, 22, 31, 22, 10, 1, 4, 22, 82, 212, 374, 450, 342, 156, 36, 2, 5, 41, 231, 955, 2880, 6459, 10660, 12948, 11274, 6802, 2645, 595, 57, 2, 7, 72, 566, 3335, 14883, 51470, 139224, 297048, 500147, 661796, 681101, 536322, 314753, 132490

Offset: 1

Author

Heinrich Ludwig, Feb 11 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 1 to A007980(n - 1).

Column #1 (k = 1) is A001399.

Column #2 (k = 2) is A227327.

Without the restriction "non-equivalent (mod D_3)" numbers are given by A234251.

Examples

			Triangle begins
1;
1,  1;
2,  4,   4,   2;
3, 10,  22,  31,   22,   10,     1;
4, 22,  82, 212,  374,  450,   342,   156,    36,    2;
5, 41, 231, 955, 2880, 6459, 10660, 12948, 11274, 6802, 2645, 595, 57, 2;
...
There are exactly T(5, 10) = 2 non-equivalent ways to choose 10 points (X) from a triangular grid of side 5 avoiding that any three of them form a subtriangle of side 2.
       .                X
      X X              . X
     X . X            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..123

Crossrefs

Cf. A234251, A007980, A001399, A227327.