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A193127 Numbers of spanning trees of the antiprism graphs. 0

%I

%S 2,36,384,3528,30250,248832,1989806,15586704,120187008,915304500,

%T 6900949462,51599794176,383142771674,2828107288188,20768716848000,

%U 151840963183392,1105779284582146,8024954790380544,58059628319357318,418891171182561000

%N Numbers of spanning trees of the antiprism graphs.

%C Antiprism graphs are defined for n>=3; extended to n=1 using closed form.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (16, -80, 130, -80, 16, -1).

%F a(n) = 2/5*n*(phi^(4*n) + phi^(-4*n) - 2), where phi is the golden ratio.

%F a(n) = +16*a(n-1)-80*a(n-2)+130*a(n-3)-80*a(n-4)+16*a(n-5)-a(n-6).

%F O.g.f.: (2*x*(1 + 2*x - 16*x^2 + 2*x^3 + x^4))/((-1 + x)^2*(1 - 7*x + x^2)^2).

%F 5*a(n) = 2*n*(A056854(n) - 2). - _Eric W. Weisstein_, Mar 28 2018

%t Table[2 n (GoldenRatio^(4 n) + GoldenRatio^(-4 n) - 2)/5, {n, 20}] // Round

%t LinearRecurrence[{16, -80, 130, -80, 16, -1}, {2, 36, 384, 3528, 30250, 248832}, 20]

%t CoefficientList[Series[(2 (1 + 2 x - 16 x^2 + 2 x^3 + x^4))/((-1 + x)^2 (1 - 7 x + x^2)^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Mar 28 2018 *)

%t Table[2 n (LucasL[4 n] - 2)/5, {n, 20}] (* _Eric W. Weisstein_, Mar 28 2018 *)

%o (PARI) a(n)=my(x=quadgen(5)^n); real(2*n*(x^4+x^-4-2)/5) \\ _Charles R Greathouse IV_, Dec 17 2013

%K nonn,easy

%O 1,1

%A _Eric W. Weisstein_, Jul 16 2011

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Last modified August 8 09:27 EDT 2022. Contains 356005 sequences. (Running on oeis4.)