|
%I #33 Feb 08 2023 09:34:55
%S 0,0,0,10,40,260,1240,6510,32240,162760,812240,4069010,20337240,
%T 101725260,508587240,2543131510,12715462240,63578287760,317890462240,
%U 1589457194010,7947281087240,39736429850260
%N Number of walks of length n along the edges of an icosahedron between two opposite vertices.
%H Vincenzo Librandi, <a href="/A054885/b054885.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,10,-20,-25).
%F G.f.: (1/12)*(1/(1-5*t) + 5/(1+t) - 6/(1-5*t^2)).
%F a(n) = (5^n + 5*(-1)^n - 3*(1 + (-1)^n)*sqrt(5)^n)/12.
%F a(n+1) = 5 * A030518(n) for n > 0.
%F a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4). - _François Marques_, Jul 10 2021
%F E.g.f.: (1/12)*(5*exp(-x) + exp(5*x) - 6*cosh(sqrt(5)*x)). - _G. C. Greubel_, Feb 07 2023
%t LinearRecurrence[{4,10,-20,-25}, {0,0,0,10}, 41] (* _G. C. Greubel_, Feb 07 2023 *)
%o (Magma) [Floor((5^n+(-1)^n*5-3*(1+(-1)^n)*Sqrt(5)^n)/12): n in [0..30]]; // _Vincenzo Librandi_, Aug 24 2011
%o (PARI) a(n) = if(n%2, 5^n-5, 5^n+5-6*5^(n/2))/12; \\ _François Marques_, Jul 11 2021
%o (SageMath)
%o def A054885(n): return (5^n +5*(-1)^n -3*(1+(-1)^n)*5^(n/2))/12
%o [A054885(n) for n in range(41)] # _G. C. Greubel_, Feb 07 2023
%Y Cf. A030517, A030518, A054881, A054882, A054883, A054884.
%K nonn,walk,easy
%O 0,4
%A Paolo Dominici (pl.dm(AT)libero.it), May 23 2000
|
