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A350816 Number of minimum dominating sets in the 2 X n king graph.

%I A350816 #9 Feb 16 2025 08:34:02
%S A350816 2,4,2,16,12,4,64,32,8,208,80,16,608,192,32,1664,448,64,4352,1024,128,
%T A350816 11008,2304,256,27136,5120,512,65536,11264,1024,155648,24576,2048,
%U A350816 364544,53248,4096,843776,114688,8192,1933312,245760,16384,4390912,524288,32768
%N A350816 Number of minimum dominating sets in the 2 X n king graph.
%H A350816 Andrew Howroyd, <a href="/A350816/b350816.txt">Table of n, a(n) for n = 1..1000</a>
%H A350816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%H A350816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumDominatingSet.html">Minimum Dominating Set</a>
%H A350816 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-12,0,0,8).
%F A350816 a(n) = 6*a(n-3) - 12*a(n-6) + 8*a(n-9) for n > 9.
%F A350816 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.
%F A350816 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).
%F A350816 a(3*n) = A000079(n); a(3*n+1) = A076616(n+3); a(3*n+2) = A001787(n+2).
%o A350816 (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))
%o A350816 (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))}
%Y A350816 Row 2 of A350815.
%Y A350816 Cf. A000079, A001787, A076616, A347558, A347633.
%K A350816 nonn,easy
%O A350816 1,1
%A A350816 _Andrew Howroyd_, Jan 17 2022