This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193127 #27 Feb 16 2025 08:33:15
%S A193127 2,36,384,3528,30250,248832,1989806,15586704,120187008,915304500,
%T A193127 6900949462,51599794176,383142771674,2828107288188,20768716848000,
%U A193127 151840963183392,1105779284582146,8024954790380544,58059628319357318,418891171182561000,3014678940049375872,21646865272061272716
%N A193127 Numbers of spanning trees of the antiprism graphs.
%C A193127 Antiprism graphs are defined for n>=3; extended to n=1 using closed form.
%H A193127 Stefano Spezia, <a href="/A193127/b193127.txt">Table of n, a(n) for n = 1..1100</a>
%H A193127 Zbigniew R. Bogdanowicz, <a href="https://www.dmlett.com/archive/v13/DML24_v13_pp66-73.pdf">The number of spanning trees in a superprism</a>, Discrete Math. Lett. 13 (2024) 66-73. See p. 66.
%H A193127 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H A193127 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%H A193127 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (16,-80,130,-80,16,-1).
%F A193127 a(n) = 2/5*n*(phi^(4*n) + phi^(-4*n) - 2), where phi is the golden ratio.
%F A193127 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 A193127 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 A193127 5*a(n) = 2*n*(A056854(n) - 2). - _Eric W. Weisstein_, Mar 28 2018
%t A193127 Table[2 n (GoldenRatio^(4 n) + GoldenRatio^(-4 n) - 2)/5, {n, 20}] // Round
%t A193127 LinearRecurrence[{16, -80, 130, -80, 16, -1}, {2, 36, 384, 3528, 30250, 248832}, 20]
%t A193127 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 A193127 Table[2 n (LucasL[4 n] - 2)/5, {n, 20}] (* _Eric W. Weisstein_, Mar 28 2018 *)
%o A193127 (PARI) a(n)=my(x=quadgen(5)^n); real(2*n*(x^4+x^-4-2)/5) \\ _Charles R Greathouse IV_, Dec 17 2013
%Y A193127 Cf. A056854.
%K A193127 nonn,easy
%O A193127 1,1
%A A193127 _Eric W. Weisstein_, Jul 16 2011