A362016 Maximal number of unmarked cells with at least 3 marked neighboring cells in the n X n kings' graph.

0, 1, 4, 8, 13, 20, 28, 38, 50, 61, 75, 90, 108, 124, 139

Offset: 1

Author

Tomas Rigaux, Apr 04 2023

Keywords

• nonn
• hard
• more

Comments

The value of r = lim sup a(n) / n^2 is in the half-open interval [2/3, 8/11).

It appears from the computed terms that r = 2/3.

Examples

			a(2) = 1, as the only pattern is
  .X
  XX
a(9) = 50, with a similar pattern to prove that r >= 2/3:
  X.......X
  .XXXXXXX.
  X.......X
  .........
  XXXXXXXXX
  .........
  X.......X
  .XXXXXXX.
  X.......X
a(10) = 61, and a pattern that reaches that is
  X..X...X..
  XX.X.X.X.X
  ..........
  .X.XX.X.XX
  XX....X...
  ....X....X
  X.XX..XX.X
  X....X....
  ...X....XX
  XX.X.XX.X.
		

Links

• Tomas Rigaux, Python program

Crossrefs

If we only want 1 marked neighbor, we get n^2 - A075561(n).