A094233 Number of closed walks of length n at a vertex of the cyclic graph on 9 nodes C_9.

1, 0, 2, 0, 6, 0, 20, 0, 70, 2, 252, 22, 924, 156, 3432, 910, 12870, 4760, 48622, 23256, 184796, 108528, 705894, 490314, 2708204, 2163150, 10430500, 9373652, 40313160, 40060078, 156305070, 169345560, 607812102, 709645552, 2369918628, 2952780320

Offset: 0

Author

Herbert Kociemba, May 29 2004

Keywords

• nonn
• easy

Comments

In general, a(n,m) = (2^n/m)*Sum_{k=0..m-1} cos(2*Pi*k/m)^n gives the number of closed walks of length n at a vertex of the cyclic graph on m nodes C_m.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Crossrefs

Cf. A078008, A054877, A047849.

Programs

Mathematica

f[n_] := FullSimplify[ TrigToExp[ 2^n/9 Sum[ Cos[2Pi*k/9]^n, {k, 0, 8}]]]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v, Jun 01 2004 *)

Formula

a(n) = (2^n/9)*Sum_{k=0..8} cos(2*Pi*k/9)^n.

G.f.: -(x-1)*(x^3+3*x^2-1)/((2*x-1)*(x+1)*(x^3-3*x^2+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

9*a(n) = 2*(-1)^n +2^n +6*(-1)^n*A188048(n). - R. J. Mathar, Nov 03 2020

Extensions

More terms from Robert G. Wilson v, Jun 01 2004