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A350816
Number of minimum dominating sets in the 2 X n king graph.
4
2, 4, 2, 16, 12, 4, 64, 32, 8, 208, 80, 16, 608, 192, 32, 1664, 448, 64, 4352, 1024, 128, 11008, 2304, 256, 27136, 5120, 512, 65536, 11264, 1024, 155648, 24576, 2048, 364544, 53248, 4096, 843776, 114688, 8192, 1933312, 245760, 16384, 4390912, 524288, 32768
OFFSET
1,1
LINKS
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Minimum Dominating Set
FORMULA
a(n) = 6*a(n-3) - 12*a(n-6) + 8*a(n-9) for n > 9.
G.f.: 2*x*(1 - x)*(1 + 3*x + 4*x^2 + 6*x^3 - 4*x^5 - 8*x^6 - 4*x^7)/(1 - 2*x^3)^3.
a(3*n) = 2^n; a(3*n+1) = (n^2 + 5*n + 2)*2^n; a(3*n+2) = (n + 2)*2^(n+1).
a(3*n) = A000079(n); a(3*n+1) = A076616(n+3); a(3*n+2) = A001787(n+2).
PROG
(PARI) Vec(2*(1 - x)*(1 + 3*x + 4*x^2 + 6*x^3 - 4*x^5 - 8*x^6 - 4*x^7)/(1 - 2*x^3)^3 + O(x^45))
(PARI) a(n) = {my(t=n\3); 2^t*if(n%3==0, 1, if(n%3==1, t^2 + 5*t + 2, 2*t + 4))}
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Andrew Howroyd, Jan 17 2022
STATUS
approved