A362016 - OEIS

%I #18 Aug 24 2023 09:43:37

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

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

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

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

%H Tomas Rigaux, <a href="/A362016/a362016.py.txt">Python program</a>

%e a(2) = 1, as the only pattern is

%e   .X

%e   XX

%e a(9) = 50, with a similar pattern to prove that r >= 2/3:

%e   X.......X

%e   .XXXXXXX.

%e   X.......X

%e   .........

%e   XXXXXXXXX

%e   .........

%e   X.......X

%e   .XXXXXXX.

%e   X.......X

%e a(10) = 61, and a pattern that reaches that is

%e   X..X...X..

%e   XX.X.X.X.X

%e   ..........

%e   .X.XX.X.XX

%e   XX....X...

%e   ....X....X

%e   X.XX..XX.X

%e   X....X....

%e   ...X....XX

%e   XX.X.XX.X.

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

%K nonn,hard,more

%O 1,3

%A _Tomas Rigaux_, Apr 04 2023