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A350817 Number of minimum total dominating sets in the 2 X n king graph.

%I A350817 #13 May 29 2025 10:40:46
%S A350817 1,6,9,4,8,89,56,16,64,780,304,64,384,5472,1536,256,2048,33920,7424,
%T A350817 1024,10240,194304,34816,4096,49152,1053696,159744,16384,229376,
%U A350817 5488640,720896,65536,1048576,27721728,3211264,262144,4718592,136642560,14155776,1048576
%N A350817 Number of minimum total dominating sets in the 2 X n king graph.
%H A350817 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>.
%H A350817 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumTotalDominatingSet.html">Minimum Total Dominating Set</a>.
%H A350817 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,12,0,0,0,-48,0,0,0,64).
%F A350817 a(n) = 12*a(n-4) - 48*a(n-8) + 64*a(n-12) for n > 13.
%F A350817 G.f.: x*(1 + 6*x + 9*x^2 + 4*x^3 - 4*x^4 + 17*x^5 - 52*x^6 - 32*x^7 + 16*x^8 + 64*x^10 + 64*x^11 - 64*x^12)/((1 - 2*x^2)^3*(1 + 2*x^2)^3).
%F A350817 a(4*k) = 4^k; a(4*k+1) = 2*k*4^k for k > 0; a(4*k+2) = (k + 1)*(41*k + 48)*4^k/8; a(4*k+3) = (5*k + 9)*4^k.
%t A350817 LinearRecurrence[{0, 0, 0, 12, 0, 0, 0, -48, 0, 0, 0, 64}, {1, 6, 9, 4, 8, 89, 56, 16, 64, 780, 304, 64, 384}, 40] (* _Michael De Vlieger_, Jan 19 2022 *)
%o A350817 (PARI) Vec((1 + 6*x + 9*x^2 + 4*x^3 - 4*x^4 + 17*x^5 - 52*x^6 - 32*x^7 + 16*x^8 + 64*x^10 + 64*x^11 - 64*x^12)/((1 - 2*x^2)^3*(1 + 2*x^2)^3) + O(x^40))
%o A350817 (PARI) a(n)={my(k=n\4); 4^k*if(n%2, if(n%4==1, (k==0) + 2*k, 5*k + 9), if(n%4==0, 1, (k + 1)*(41*k + 48)/8))}
%Y A350817 Row 2 of A303335.
%Y A350817 Cf. A350816.
%K A350817 nonn,easy
%O A350817 1,2
%A A350817 _Andrew Howroyd_, Jan 17 2022