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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A340755 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

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Last modified October 21 05:28 EDT 2021. Contains 348141 sequences. (Running on oeis4.)