A284647 Number of nonisomorphic unfoldings in an n-gonal Archimedean antiprism.
0, 1, 6, 36, 231, 1540, 10440, 71253, 487578, 3339820, 22885995, 156848616, 1075018896, 7368190921, 50502074766, 346145696820, 2372516138895, 16261462918828, 111457712887128, 763942497430365, 5236139690949090, 35889035134544956, 245987105715037011
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Takashi Horiyama and Wataru Shoji, The Number of Different Unfoldings of Polyhedra. In: L. Cai, S.-W. Cheng, and T.-W. Lam (Eds.): ISAAC2013, LNCS 8283, pp. 623-633, Springer-Verlag, 2013.
- Rick Mabry, Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms, The American Mathematical Monthly, Vol. 126 (2019), no. 9, pp. 786-801.
- Index entries for linear recurrences with constant coefficients, signature (11,-33,33,-11,1).
Programs
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Maple
a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 36, 231][n+1], 11*(a(n-1)-3*(a(n-2)-a(n-3))-a(n-4))+a(n-5)) end: seq(a(n), n=0..30); # Alois P. Heinz, Mar 31 2017 -
Mathematica
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 *) LinearRecurrence[{11,-33,33,-11,1},{0,1,6,36,231},30] (* Harvey P. Dale, Jan 07 2021 *) -
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
Formula
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)).
From Colin Barker, Mar 31 2017: (Start)
G.f.: x*(1 - 5*x + 3*x^2) / ((1 - x)*(1 - 7*x + x^2)*(1 - 3*x + x^2)).
a(n) = 11*a(n-1) - 33*a(n-2) + 33*a(n-3) - 11*a(n-4) - a(n-5) for n>4.
(End)
a(n) = (5*Fibonacci(2*n) + Lucas(4*n) - 2)/10. - Ehren Metcalfe, Apr 21 2018
a(n) = Fibonacci(2*n)*(1+Fibonacci(2*n))/2 - Rick Mabry, Apr 10 2021