A054884 Number of closed walks of length n along the edges of an icosahedron based at a vertex.

1, 0, 5, 10, 65, 260, 1365, 6510, 32865, 162760, 815365, 4069010, 20352865, 101725260, 508665365, 2543131510, 12715852865, 63578287760, 317892415365, 1589457194010, 7947290852865, 39736429850260

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,10,-20,-25).

Formula

G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) + 6/(1-5*t^2)).

a(n) = (5^n + (-1)^n*5 + 3*(1 + (-1)^n)*sqrt(5)^n)/12.

a(n+1) = 5 * A030517(n) for n > 0.

a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4). - François Marques, Jul 10 2021

E.g.f.: (1/12)*(5*exp(-x) + exp(5*x) + 6*cosh(sqrt(5)*x)). - G. C. Greubel, Feb 07 2023

Mathematica

LinearRecurrence[{4, 10, -20, -25}, {1, 0, 5, 10}, 30] (* Harvey P. Dale, May 02 2022 *)

Prog

(Magma) [Floor((5^n+(-1)^n*5+3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011
(PARI) a(n) = if(n%2, 5^n-5, 5^n+5+6*5^(n/2))/12; \\ François Marques, Jul 11 2021
(SageMath)
def A054884(n): return (5^n + 5*(-1)^n + 3*(1 + (-1)^n)*5^(n/2))/12
[A054884(n) for n in range(41)] # G. C. Greubel, Feb 07 2023

Crossrefs

Cf. A030517, A030518, A054881, A054882, A054883, A054885.

Sequence in context: A240647 A062162 A062848 * A061518 A218540 A174937

Adjacent sequences: A054881 A054882 A054883 * A054885 A054886 A054887

Keyword

nonn,walk,easy

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Status

approved