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

1,4

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).

Links

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

Example

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

Crossrefs

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

Sequence in context: A125941 A347270 A275117 * A243207 A297003 A071284

Adjacent sequences: A243138 A243139 A243140 * A243142 A243143 A243144

Keyword

nonn,tabf

Author

Heinrich Ludwig, May 30 2014

Status

approved