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

1,5

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.

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..152

Example

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 . .

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.

Sequence in context: A369772 A059215 A125142 * A324035 A096352 A260720

Adjacent sequences: A234347 A234348 A234349 * A234351 A234352 A234353

Keyword

nonn,tabf,nice

Author

Heinrich Ludwig, Dec 24 2013

Status

approved