A094233 - OEIS

%I #18 Aug 28 2024 04:26:03

%S 1,0,2,0,6,0,20,0,70,2,252,22,924,156,3432,910,12870,4760,48622,23256,

%T 184796,108528,705894,490314,2708204,2163150,10430500,9373652,

%U 40313160,40060078,156305070,169345560,607812102,709645552,2369918628,2952780320

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

%C 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.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-4,-5,2).

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

%F 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

%F 9*a(n) = 2*(-1)^n +2^n +6*(-1)^n*A188048(n). - _R. J. Mathar_, Nov 03 2020

%t 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 *)

%Y Cf. A078008, A054877, A047849.

%K nonn,easy

%O 0,3

%A _Herbert Kociemba_, May 29 2004

%E More terms from _Robert G. Wilson v_, Jun 01 2004