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A193127
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Numbers of spanning trees of the antiprism graphs.
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0
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2, 36, 384, 3528, 30250, 248832, 1989806, 15586704, 120187008, 915304500, 6900949462, 51599794176, 383142771674, 2828107288188, 20768716848000, 151840963183392, 1105779284582146, 8024954790380544, 58059628319357318, 418891171182561000
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OFFSET
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1,1
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COMMENTS
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Antiprism graphs are defined for n>=3; extended to n=1 using closed form.
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LINKS
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Table of n, a(n) for n=1..20.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Index entries for linear recurrences with constant coefficients, signature (16, -80, 130, -80, 16, -1).
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FORMULA
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a(n) = 2/5*n*(phi^(4*n) + phi^(-4*n) - 2), where phi is the golden ratio.
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).
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).
5*a(n) = 2*n*(A056854(n) - 2). - Eric W. Weisstein, Mar 28 2018
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MATHEMATICA
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Table[2 n (GoldenRatio^(4 n) + GoldenRatio^(-4 n) - 2)/5, {n, 20}] // Round
LinearRecurrence[{16, -80, 130, -80, 16, -1}, {2, 36, 384, 3528, 30250, 248832}, 20]
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 *)
Table[2 n (LucasL[4 n] - 2)/5, {n, 20}] (* Eric W. Weisstein, Mar 28 2018 *)
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PROG
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(PARI) a(n)=my(x=quadgen(5)^n); real(2*n*(x^4+x^-4-2)/5) \\ Charles R Greathouse IV, Dec 17 2013
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CROSSREFS
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Sequence in context: A157055 A057407 A286266 * A092852 A139738 A248343
Adjacent sequences: A193124 A193125 A193126 * A193128 A193129 A193130
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KEYWORD
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nonn,easy
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AUTHOR
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Eric W. Weisstein, Jul 16 2011
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STATUS
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approved
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