A054885 Number of walks of length n along the edges of an icosahedron between two opposite vertices.
0, 0, 0, 10, 40, 260, 1240, 6510, 32240, 162760, 812240, 4069010, 20337240, 101725260, 508587240, 2543131510, 12715462240, 63578287760, 317890462240, 1589457194010, 7947281087240, 39736429850260
Offset: 0
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).
Programs
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Magma
[Floor((5^n+(-1)^n*5-3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // Vincenzo Librandi, Aug 24 2011
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Mathematica
LinearRecurrence[{4,10,-20,-25}, {0,0,0,10}, 41] (* G. C. Greubel, Feb 07 2023 *) -
PARI
a(n) = if(n%2, 5^n-5, 5^n+5-6*5^(n/2))/12; \\ François Marques, Jul 11 2021
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SageMath
def A054885(n): return (5^n +5*(-1)^n -3*(1+(-1)^n)*5^(n/2))/12 [A054885(n) for n in range(41)] # G. C. Greubel, Feb 07 2023
Formula
G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) - 6/(1-5*t^2)).
a(n) = (5^n + 5*(-1)^n - 3*(1 + (-1)^n)*sqrt(5)^n)/12.
a(n+1) = 5 * A030518(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