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%I #43 Apr 10 2021 15:47:51
%S 0,1,6,36,231,1540,10440,71253,487578,3339820,22885995,156848616,
%T 1075018896,7368190921,50502074766,346145696820,2372516138895,
%U 16261462918828,111457712887128,763942497430365,5236139690949090,35889035134544956,245987105715037011
%N Number of nonisomorphic unfoldings in an n-gonal Archimedean antiprism.
%H Colin Barker, <a href="/A284647/b284647.txt">Table of n, a(n) for n = 0..1000</a>
%H Takashi Horiyama and Wataru Shoji, <a href="http://doi.org/10.1007/978-3-642-45030-3_58">The Number of Different Unfoldings of Polyhedra</a>. In: L. Cai, S.-W. Cheng, and T.-W. Lam (Eds.): ISAAC2013, LNCS 8283, pp. 623-633, Springer-Verlag, 2013.
%H Rick Mabry, <a href="https://doi.org/10.1080/00029890.2019.1644124">Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms</a>, The American Mathematical Monthly, Vol. 126 (2019), no. 9, pp. 786-801.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (11,-33,33,-11,1).
%F a(n) = (1/10)*( ((1 + sqrt(5))/2)^(4*n) + ((1 + sqrt(5))/2)^(-4*n) - 2) + ((3 + sqrt(5))^n - (3 - sqrt(5))^n )/(2^(n + 1)*sqrt(5)).
%F From _Colin Barker_, Mar 31 2017: (Start)
%F G.f.: x*(1 - 5*x + 3*x^2) / ((1 - x)*(1 - 7*x + x^2)*(1 - 3*x + x^2)).
%F a(n) = 11*a(n-1) - 33*a(n-2) + 33*a(n-3) - 11*a(n-4) - a(n-5) for n>4.
%F (End)
%F a(n) = (5*Fibonacci(2*n) + Lucas(4*n) - 2)/10. - _Ehren Metcalfe_, Apr 21 2018
%F a(n) = Fibonacci(2*n)*(1+Fibonacci(2*n))/2 - _Rick Mabry_, Apr 10 2021
%p a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 36, 231][n+1],
%p 11*(a(n-1)-3*(a(n-2)-a(n-3))-a(n-4))+a(n-5))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 31 2017
%t CoefficientList[Series[x(1 - 5x + 3x^2) / ((1 - x)*(1 - 7x + x^2)*(1 - 3x + x^2)), {x, 0, 25}], x] (* _Indranil Ghosh_, Mar 31 2017 *)
%t LinearRecurrence[{11,-33,33,-11,1},{0,1,6,36,231},30] (* _Harvey P. Dale_, Jan 07 2021 *)
%o (PARI) concat(0, Vec(x*(1 - 5*x + 3*x^2) / ((1 - x)*(1 - 7*x + x^2)*(1 - 3*x + x^2)) + O(x^30))) \\ _Colin Barker_, Mar 31 2017
%K nonn,easy
%O 0,3
%A _Humberto Bortolossi_, Mar 31 2017
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