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

Author

Eric W. Weisstein, Jul 16 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A056854.

Programs

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 *)

PARI

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

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