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%I #22 Mar 03 2024 23:50:47
%S 2,56,396,2040,9130,37944,151172,586608,2235618,8407640,31292844,
%T 115494312,423283562,1542120664,5589611460,20170172896,72499928322,
%U 259692909048,927342338956,3302291258200,11730149911914,41572470711288,147031327493572,519029653663056
%N Number of (undirected) paths in the n-antiprism graph.
%C Sequence extrapolated to n=1 using recurrence. - _Andrew Howroyd_, Jun 05 2017
%H Andrew Howroyd, <a href="/A287988/b287988.txt">Table of n, a(n) for n = 1..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphPath.html">Graph Path</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10, -37, 64, -58, 36, -26, 16, -5, 2, -1).
%F From _Andrew Howroyd_, Jun 05 2017 (Start)
%F a(n) = 10*a(n-1)-37*a(n-2)+64*a(n-3) -58*a(n-4)+36*a(n-5)-26*a(n-6) +16*a(n-7)-5*a(n-8) +2*a(n-9)-a(n-10) for n>10.
%F G.f.: 2*x*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1) * (1-2*x-x^2) / ((1-x)^4 * (1-3*x-x^2-x^3)^2).
%F (End)
%t Table[n RootSum[-1 - # - 3 #^2 + #^3 &, 23 #^n + 32 #^(n + 1) + 5 #^(n + 2) &]/44 - 7 n - 3 n^2 - 2 n^3, {n, 20}]
%t LinearRecurrence[{10, -37, 64, -58, 36, -26, 16, -5, 2, -1}, {2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640}, 20]
%t CoefficientList[Series[(2 (2 x^6 + 4 x^5 + x^4 + 24 x^3 - 4 x^2 + 20 x + 1) (1 - 2 x - x^2))/((1 - x)^4 (1 - 3 x - x^2 - x^3)^2), {x, 0, 20}], x]
%o (PARI)
%o Vec(2*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1)*(1-2*x-x^2)/((1-x)^4*(1-3*x-x^2-x^3)^2) + O(x^20)) \\ _Andrew Howroyd_, Jun 05 2017
%Y Cf. A077263, A124352, A124353, A287992.
%K nonn
%O 1,1
%A _Eric W. Weisstein_, Jun 03 2017
%E a(1)-a(2) and a(14)-a(24) from _Andrew Howroyd_, Jun 05 2017
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