This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A244506 #17 Feb 16 2025 08:33:23 %S A244506 1,9,196,6084,219024,8450649,338265664,13840346025,574510941225, %T A244506 24093764931600 %N 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). %C A244506 (1) All a(n) are square numbers. The sequence of their roots is A244507. %C A244506 (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). %C A244506 (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): %C A244506 X %C A244506 . . %C A244506 . . . %C A244506 x . . X %C A244506 . . . . . %C A244506 . . . . . . %C A244506 X . . X . . X %C A244506 (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). %H A244506 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumIndependentVertexSet.html">Maximum Independent Vertex Set</a> %Y A244506 Cf. A000326, A244507, A244500. %Y A244506 Cf. A297537 (maximum independent vertex sets for n-triangular honeycomb acute knight graph). %K A244506 nonn,hard,more %O A244506 1,2 %A A244506 _Heinrich Ludwig_, Jul 11 2014 %E A244506 a(10) from _Heinrich Ludwig_, Aug 17 2014