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A383420 Maximum (equal) number of red and blue tiles on an n X n matrix, where opposite colors cannot be adjacent diagonally or edgewise, and where a cluster of the same color can be no greater than n.

%I A383420 #43 May 04 2025 12:54:20
%S A383420 0,0,6,8,16,24,30,38
%N A383420 Maximum (equal) number of red and blue tiles on an n X n matrix, where opposite colors cannot be adjacent diagonally or edgewise, and where a cluster of the same color can be no greater than n.
%C A383420 a(n) refers to the total number of both red and blue tiles covering the n X n matrix, and thus all the terms are even.
%H A383420 Jakub Buczak, <a href="/A383420/a383420_3.txt">Optimal matrix example with a(n) tiles for each n</a>
%H A383420 Jakub Buczak, <a href="/A383420/a383420_1.jpg">All 105 arrangements for n=4 with 8 tiles ignoring rotation, mirroring and color swapping.</a>
%F A383420 The lower and upper bounds apply to all n > 3.
%F A383420 k is the root of the smallest square greater than n, b = ceiling((n+1)/(k+1)). The variable x is defined as 5*((n+1)/(k+1) - b), if (n+1)/(k+1) - b > 0, otherwise x=0.
%F A383420 a(n) <= (b*k)^2 + n^2 - (n - x)^2 for all n.
%F A383420 a(n) => n^2/2, for n == 0 mod 4.
%F A383420 a(n) => (n^2 + n - 2)/2, for n == 1 mod 4.
%F A383420 a(n) => n^2/2 + n - 4, for n == 2 mod 4.
%F A383420 a(n) => (n^2 + n - 4)/2, for n == 3 mod 4.
%e A383420 a(2) = 0, since red and blue tiles, regardless of the arrangement will always either border sideways or diagonally.
%e A383420 a(3) = 6, a simple example could be:
%e A383420   [R R R]
%e A383420   [     ]
%e A383420   [B B B]
%Y A383420 Cf. A000290, A002378, A023365, A001105, A033587.
%K A383420 nonn,hard,more
%O A383420 1,3
%A A383420 _Jakub Buczak_, Apr 26 2025