A234350 Triangle T(n, k) = Number of non-equivalent (mod D_3) ways to arrange k indistinguishable points on a triangular grid of side n so that no three points are collinear. Triangle read by rows.

1, 1, 1, 1, 2, 4, 5, 2, 3, 10, 22, 24, 8, 1, 4, 22, 77, 153, 140, 47, 2, 5, 41, 217, 713, 1290, 1112, 322, 15, 7, 72, 530, 2557, 7374, 11743, 8783, 2412, 143, 1, 8, 116, 1149, 7661, 32477, 82988, 116154, 77690, 19621, 1220, 5, 10, 180, 2288, 20055, 116420, 433372

Offset: 1

Author

Heinrich Ludwig, Dec 24 2013

Keywords

• nonn
• tabf
• nice

Comments

The triangle T(n, k) is irregularly shaped: 1 <= k <= A234349(n). First row corresponds to n = 1.

The maximal number of points that can be placed on a triangular grid of side n so that no three points are collinear is given by A234349(n).

Without the restriction "non-equivalent (mod D_3)" the numbers are given by A194136.

Examples

			Triangle begins
1;
1,   1,    1;
2,   4,    5,    2;
3,  10,   22,   24,     8,     1;
4,  22,   77,  153,   140,    47,      2;
5,  41,  217,  713,  1290,  1112,    322,    15;
7,  72,  530, 2557,  7374, 11743,   8783,  2412,   143,    1;
8, 116, 1149, 7661, 32477, 82988, 116154, 77690, 19621, 1220, 5;
...
There are e.g. T(8, 11) = 5 non-equivalent ways to arrange 11 indistinguishable points (X) on a triangular grid of side 8 so that no point triple is collinear. As examples of the 5 solutions the 2 symmetrical ones are shown.
          .                    .
         . .                  . .
        . X .                . X .
       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..152

Crossrefs

Cf. A194136

Row lengths are given by A234349

Column 1 is A001399

Column 2 is A227327 for n >= 2

Column 3 is A234351

Column 4 is A234352

Column 5 is A234353

Column 6 is A234354.