A297537 -id:A297537 - cp's OEIS frontend

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

Heinrich Ludwig, Jul 11 2014

(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).

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

Cf. A000326, A244507, A244500.

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

a(10) from Heinrich Ludwig, Aug 17 2014

cp's OEIS Frontend

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).

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