A243207 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 with sides parallel to the grid. Triangle read by rows.

1, 1, 1, 2, 4, 3, 1, 3, 10, 20, 25, 11, 3, 4, 22, 77, 186, 266, 221, 86, 14, 5, 41, 223, 881, 2344, 4238, 4885, 3451, 1296, 220, 7, 1, 7, 72, 552, 3146, 12907, 38640, 83107, 126701, 132236, 90214, 37128, 8235, 775, 24, 8, 116, 1196, 9264, 53307, 232861, 773930

Offset

1,4

Comments

The triangle T(n, k) is irregularly shaped: 1 <= k <= A227308(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 with sides parallel to the grid is given by A227308(n).

Links

Heinrich Ludwig, Table of n, a(n) for n = 1..153

Example

The triangle begins:
  1;
  1,  1;
  2,  4,   3,   1;
  3, 10,  20,  25,   11,    3;
  4, 22,  77, 186,  266,  221,   86,   14;
  5, 41, 223, 881, 2344, 4238, 4885, 3451, 1296, 220, 7, 1;
  ...
There is T(6, 12) = 1 way to place 12 points (x) on the grid obeying the rule in the definition of the sequence:
           .
          x x
         x . x
        x . . x
       x . . . x
      . x x x x .

Crossrefs

Cf. A227308, A243211, A239572, A234247, A231655, A243141, A001399 (column 1), A227327 (column 2), A243208 (column 3), A243209 (column 4), A243210 (column 5).

Sequence in context: A347270 A275117 A243141 * A297003 A071284 A104753

Adjacent sequences: A243204 A243205 A243206 * A243208 A243209 A243210

Keyword

tabf,nonn

Author

Heinrich Ludwig, Jun 01 2014

Status

approved