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

1,4

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 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.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..123

Example

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

Crossrefs

Cf. A234251, A007980, A001399, A227327.

Sequence in context: A256066 A096832 A016588 * A160163 A263442 A206396

Adjacent sequences: A234244 A234245 A234246 * A234248 A234249 A234250

Keyword

nonn,tabf

Author

Heinrich Ludwig, Feb 11 2014

Status

approved