A286850 - OEIS

%I #22 Feb 16 2025 08:33:45

%S 2,4,6,16,20,52,80,176,296,592,1104,2064,3936,7296,14048,25984,49600,

%T 92736,175872,330240,623232,1175296,2213632,4176128,7863808,14838784,

%U 27948544,52707328,99320832,187257856,352940032,665276416,1254090752,2363805696,4455927808

%N Number of minimal dominating sets in the 2 X n king graph.

%H Andrew Howroyd, <a href="/A286850/b286850.txt">Table of n, a(n) for n = 1..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimalDominatingSet.html">Minimal Dominating Set</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 2, 2, 4, 0, -8).

%F a(n) = 2*a(n-2)+2*a(n-3)+4*a(n-4)-8*a(n-6) for n>6.

%F G.f.: 2*x*(1 + 2*x + x^2 + 2*x^3 - 4*x^4 - 4*x^5)/(1 - 2*x^2 - 2*x^3 - 4*x^4 + 8*x^6).

%t Table[RootSum[8 - 4 #1^2 - 2 #1^3 - 2 #1^4 + #1^6 &, 36 #1^n - 36 #1^(2 + n) + 55 #1^(3 + n) - 3 #1^(4 + n) + 32 #1^(5 + n) &]/970, {n, 10}] (* _Eric W. Weisstein_, Aug 04 2017 *)

%t LinearRecurrence[{0, 2, 2, 4, 0, -8}, {2, 4, 6, 16, 20, 52}, 20] (* _Eric W. Weisstein_, Aug 03 2017 *)

%t CoefficientList[Series[-((2 (-1 - 2 x - x^2 - 2 x^3 + 4 x^4 + 4 x^5))/(1 - 2 x^2 - 2 x^3 - 4 x^4 + 8 x^6)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 03 2017 *)

%o (PARI)

%o Vec(2*(1+2*x+x^2+2*x^3-4*x^4-4*x^5)/(1-2*x^2-2*x^3-4*x^4+8*x^6)+O(x^40))

%Y Row 2 of A286849.

%K nonn

%O 1,1

%A _Andrew Howroyd_, Aug 01 2017