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

1, 0, 2, 0, 6, 2, 20, 14, 70, 72, 254, 330, 948, 1430, 3614, 6008, 13990, 24786, 54740, 101118, 215766, 409640, 854702, 1652090, 3396916, 6643782, 13530350, 26667864, 53971350, 106914242, 215492564, 428292590, 860941798, 1714834440, 3441074654, 6863694378

Offset: 0

Author

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

Keywords

• nonn
• walk
• easy

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2).

Crossrefs

Cf. A000032, A052964.

Programs

GAP

List([0..40], n-> (2^n + 2*(-1)^n*Lucas(1,-1, n)[2])/5); # G. C. Greubel, Jul 19 2019

Magma

[(2^n + 2*(-1)^n*Lucas(n))/5: n in [0..40]]; // G. C. Greubel, Jul 19 2019

Mathematica

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 *)

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

Sage

[(2^n + 2*(-1)^n*lucas_number2(n,1,-1))/5 for n in (0..40)] # G. C. Greubel, Jul 19 2019

Formula

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

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

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

a(n) = (2^n/5)*Sum_{k=0..4} cos(2*Pi*k/5)^n. - Herbert Kociemba, May 31 2004

Recurrence: a(n) = 5*(a(n-2) - a(n-4)) + 2*a(n-5). - Herbert Kociemba, Jun 04 2004