A243141 Triangle T(n, k) = Numbers of inequivalent (mod D_3) ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle of any orientation. Triangle read by rows.

1, 1, 1, 2, 4, 3, 1, 3, 10, 19, 22, 7, 1, 4, 22, 75, 170, 204, 115, 18, 1, 5, 41, 218, 816, 1891, 2635, 1909, 628, 58, 3, 7, 72, 542, 2947, 10846, 26695, 41770, 39218, 19905, 4776, 437, 13, 8, 116, 1178, 8765, 46068, 171700, 444117, 776276, 876012, 601078, 229941

Offset: 1

Author

Heinrich Ludwig, May 30 2014

Keywords

• nonn
• tabf

Comments

The triangle T(n, k) is irregularly shaped: 1 <= k <= A240114(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 of them form an equilateral triangle is given by A240114(n).

Examples

			The triangle begins:
  1;
  1,  1;
  2,  4,   3,    1;
  3, 10,  19,   22,     7,     1;
  4, 22,  75,  170,   204,   115,    18,     1;
  5, 41, 218,  816,  1891,  2635,  1909,   628,    58,    3;
  7, 72, 542, 2947, 10846, 26695, 41770, 39218, 19905, 4776, 437, 13;
  ...
There is exactly T(5, 8) = 1 way to place 8 points (x) on a triangular grid of side 5 according to the definition of the sequence:
           .
          x x
         x . x
        x . . x
       x . . . x
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..129

Crossrefs

Cf. A240114, A240439, A001399 (column 1), A227327 (column 2), A243142 (column 3), A243143 (column 4), A243144 (column 5).