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 A054877 Closed walks of length n along the edges of a pentagon based at a vertex. 8

%I

%S 1,0,2,0,6,2,20,14,70,72,254,330,948,1430,3614,6008,13990,24786,54740,

%T 101118,215766,409640,854702,1652090,3396916,6643782,13530350,

%U 26667864,53971350,106914242,215492564,428292590,860941798

%N Closed walks of length n along the edges of a pentagon based at a vertex.

%C In general a(n,m)=2^n/m*Sum(k,0,m-1,Cos(2Pi*k/m)^n) counts closed walks of length n at a vertex of the cyclic graph on m nodes C_m. Here we have the case m=5. - _Herbert Kociemba_, May 31 2004

%H G. C. Greubel, <a href="/A054877/b054877.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = 2*A052964(n) for n>0.

%F G.f.: -1/5*1/(2*x-1)-2/5*(2+x)/(x^2-x-1).

%F a(n) = ( 2^n + 2*(-1)^n*( F(n) + F(n-2) ) )/5, for n>1, where F(n) is the n-th Fibonacci number (cf. A000045).

%F a(n) = (2^n/5)*Sum_{k=0..4} Cos(2Pi*k/5)^n). - _Herbert Kociemba_, May 31 2004

%F Recurrence: a(n) = 5*(a(n-2) - a(n-4)) + 2*a(n-5). - _Herbert Kociemba_, Jun 04 2004

%t CoefficientList[Series[-1/5*1/(2*x - 1) - 2/5*(2 + x)/(x^2 - x - 1), {x, 0, 50}], x] (* _G. C. Greubel_, Jun 07 2017 *)

%o (PARI) x='x+O('x^50); Vec(-1/5*1/(2*x-1)-2/5*(2+x)/(x^2-x-1)) \\ _G. C. Greubel_, Jun 07 2017

%Y Cf. A052964.

%K nonn,walk

%O 0,3

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

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Last modified February 19 10:24 EST 2019. Contains 320310 sequences. (Running on oeis4.)