A054877 - OEIS

%I #26 Jan 05 2025 09:23:36

%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 26667864,53971350,106914242,215492564,428292590,860941798,1714834440,3441074654,6863694378

%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>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2).

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

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

%F 5*a(n) = 2^n +2*(-1)^n*A000032(n).

%F a(n) = (2^n/5)*Sum_{k=0..4} cos(2*Pi*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/(1-2*x) +2*(2+x)/(1+x-x^2)), {x, 0, 40}], x] (* _G. C. Greubel_, Jun 07 2017 *)

%o (PARI) vector(40, n, n--; f=fibonacci; (2^n + 2*(-1)^n*(f(n+1)+f(n-1) ))/5) \\ _G. C. Greubel_, Jun 07 2017, modified Jul 19 2019

%o (Magma) [(2^n + 2*(-1)^n*Lucas(n))/5: n in [0..40]]; // _G. C. Greubel_, Jul 19 2019

%o (Sage) [(2^n + 2*(-1)^n*lucas_number2(n,1,-1))/5 for n in (0..40)] # _G. C. Greubel_, Jul 19 2019

%o (GAP) List([0..40], n-> (2^n + 2*(-1)^n*Lucas(1,-1, n)[2])/5); # _G. C. Greubel_, Jul 19 2019

%Y Cf. A000032, A052964.

%K nonn,walk,easy

%O 0,3

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