A244506 Number of ways to place the maximal number of points that can be placed on a j X j X j triangular grid, j=3n-2, so that no pair of them has distance sqrt(3).

1, 9, 196, 6084, 219024, 8450649, 338265664, 13840346025, 574510941225, 24093764931600

Offset

1,2

Comments

(1) All a(n) are square numbers. The sequence of their roots is A244507.

(2) On a j X j X j grid, j = 3n-2, the maximal number of points that can be placed is the pentagonal number A000326(n).

(3) On a maximally occupied grid, the following grid points "X" are always occupied (example for j = 7, for other j's expand this pattern):

X

. .

. . .

x . . X

. . . . .

. . . . . .

X . . X . . X

(4) For j X j X j grids, j = 3n, the corresponding numbers are cubes of Catalan numbers, A033536(n). For j = 3n-1, the corresponding numbers are always 1, A000012(n).

Links

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Crossrefs

Cf. A000326, A244507, A244500.

Cf. A297537 (maximum independent vertex sets for n-triangular honeycomb acute knight graph).

Sequence in context: A081020 A358740 A017426 * A274269 A250401 A291974

Adjacent sequences: A244503 A244504 A244505 * A244507 A244508 A244509

Keyword

nonn,hard,more

Author

Heinrich Ludwig, Jul 11 2014

Extensions

a(10) from Heinrich Ludwig, Aug 17 2014

Status

approved