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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A193560 A278390 A001498 * A243211 A199034 A138464

Adjacent sequences:  A240436 A240437 A240438 * A240440 A240441 A240442

KEYWORD

nonn,tabf

AUTHOR

Heinrich Ludwig, Apr 05 2014

STATUS

approved

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Last modified March 8 01:41 EST 2021. Contains 341934 sequences. (Running on oeis4.)