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

Author

Heinrich Ludwig, Jul 11 2014

Keywords

• nonn
• hard
• more

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

• 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).

Extensions

a(10) from Heinrich Ludwig, Aug 17 2014