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A379766 Minimum number of kings that must be placed on an n X n chessboard such that each square is attacked or occupied by at least four kings.

%I A379766 #30 Jan 27 2025 05:36:52
%S A379766 4,9,16,16,24,36,36,47,64,64,78,100,100,117,144,144,164,196,196,219,
%T A379766 256,256,282,324,324,353,400,400,432,484,484,519,576,576,614,676,676,
%U A379766 717,784,784,828,900,900,947,1024,1024,1074,1156,1156,1209,1296,1296,1352
%N A379766 Minimum number of kings that must be placed on an n X n chessboard such that each square is attacked or occupied by at least four kings.
%C A379766 At most one king can be placed on each square.
%H A379766 Dominic McCarty, <a href="/A379766/b379766.txt">Table of n, a(n) for n = 2..100</a>
%H A379766 Matthew Scroggs, <a href="https://github.com/mscroggs/oeis/blob/main/a379766.py">Python code to compute A379766</a>
%H A379766 Dominic McCarty, <a href="/A379766/a379766.txt">Java program for A379766</a>
%H A379766 Dominic McCarty, <a href="/A379766/a379766_1.txt">Illustration of a(n) for n = 2..100</a>
%H A379766 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2,-2,0,-1,1).
%F A379766 It appears that a(3n+1) = a(3n+2) - _Dominic McCarty_, Jan 17 2025
%F A379766 For n >= 2 we have a(n) = 4*floor(n/3)^2+3*floor(n/3)+2 if 3 divides n, a(n) = 4*(floor(n/3)+1)^2 otherwise. - _Benoit Cloitre_, Jan 17 2025
%F A379766 G.f.: -x^2*(4+5*x+7*x^2-2*x^4-2*x^5-8*x^3+4*x^6)/(1+x+x^2)^2/(x-1)^3 . - _R. J. Mathar_, Jan 27 2025
%e A379766 For a 5 by 5 chessboard, the sixteen kings could be placed like this:
%e A379766   kkokk
%e A379766   kkokk
%e A379766   ooooo
%e A379766   kkokk
%e A379766   kkokk
%e A379766 For a 6 by 6 chessboard, the kings could be placed like this:
%e A379766   kkookk
%e A379766   kkkkkk
%e A379766   okooko
%e A379766   okooko
%e A379766   kkkkkk
%e A379766   kkookk
%e A379766 where o is an empty square and k is a king.
%Y A379766 Cf. A379726, A379759.
%K A379766 nonn
%O A379766 2,1
%A A379766 _Matthew Scroggs_, Jan 02 2025
%E A379766 a(9)-a(100) from _Dominic McCarty_, Jan 17 2025