A193127 Numbers of spanning trees of the antiprism graphs.

2, 36, 384, 3528, 30250, 248832, 1989806, 15586704, 120187008, 915304500, 6900949462, 51599794176, 383142771674, 2828107288188, 20768716848000, 151840963183392, 1105779284582146, 8024954790380544, 58059628319357318, 418891171182561000, 3014678940049375872, 21646865272061272716

Offset

1,1

Comments

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

Links

Stefano Spezia, Table of n, a(n) for n = 1..1100

Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See p. 66.

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

Cf. A056854.

Sequence in context: A157055 A057407 A286266 * A092852 A139738 A248343

Adjacent sequences: A193124 A193125 A193126 * A193128 A193129 A193130

Keyword

nonn,easy,changed

Author

Eric W. Weisstein, Jul 16 2011

Status

approved