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A193127 Numbers of spanning trees of the antiprism graphs. 0
2, 36, 384, 3528, 30250, 248832, 1989806, 15586704, 120187008, 915304500, 6900949462, 51599794176, 383142771674, 2828107288188, 20768716848000, 151840963183392, 1105779284582146, 8024954790380544, 58059628319357318, 418891171182561000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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).

FORMULA

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Sequence in context: A157055 A057407 A286266 * A092852 A139738 A248343

Adjacent sequences:  A193124 A193125 A193126 * A193128 A193129 A193130

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein, Jul 16 2011

STATUS

approved

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Last modified July 12 13:21 EDT 2020. Contains 335663 sequences. (Running on oeis4.)