A240439 Triangle T(n, k) = Numbers of 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, 3, 3, 1, 6, 15, 15, 3, 1, 10, 45, 105, 114, 39, 3, 1, 15, 105, 420, 969, 1194, 654, 102, 3, 1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15, 1, 28, 378, 3150, 17415, 64776, 159528, 250233, 234609, 119259, 28395, 2613, 69, 1, 36, 630, 6930

Offset: 1

Author

Heinrich Ludwig, Apr 05 2014

Keywords

• nonn
• tabf

Comments

The triangle T(n, k) is irregularly shaped: 0 <= 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,  3,   3;
  1,  6,  15,   15,    3;
  1, 10,  45,  105,  114,    39,     3;
  1, 15, 105,  420,  969,  1194,   654,   102,    3;
  1, 21, 210, 1260, 4773, 11259, 15615, 11412, 3663, 342, 15;
There are T(5, 8) = 3 ways to place 8 points (x) on a triangular grid of side 5 under the conditions mentioned above:
          .                x                x
         x x              x .              . x
        x . x            x . .            . . x
       x . . x          x . . .          . . . x
      x . . . x        . x x x x        x x x x .
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..138

Crossrefs

Cf. A240114, A240443, A084546,

column 2 is A000217,

column 3 is A050534,

column 4 is A240440,

column 5 is A240441,

column 6 is A240442.