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

0,3

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).

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A244889 A344269 A144819 * A099252 A057395 A259819

Adjacent sequences: A284644 A284645 A284646 * A284648 A284649 A284650

Keyword

nonn,easy

Author

Humberto Bortolossi, Mar 31 2017

Status

approved