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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).
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
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
CROSSREFS
Cf. A297537 (maximum independent vertex sets for n-triangular honeycomb acute knight graph).
Sequence in context: A081020 A358740 A017426 * A274269 A250401 A291974
KEYWORD
nonn,hard,more
AUTHOR
Heinrich Ludwig, Jul 11 2014
EXTENSIONS
a(10) from Heinrich Ludwig, Aug 17 2014
STATUS
approved