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

1,4

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).

Links

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

Example

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 .

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.

Sequence in context: A278390 A356916 A001498 * A243211 A199034 A138464

Adjacent sequences: A240436 A240437 A240438 * A240440 A240441 A240442

Keyword

nonn,tabf

Author

Heinrich Ludwig, Apr 05 2014

Status

approved