A095364 - OEIS

%I #18 Nov 14 2024 23:23:00

%S 1,0,3,0,10,0,35,1,126,11,462,78,1716,455,6435,2380,24311,11628,92398,

%T 54264,352947,245157,1354102,1081575,5215250,4686826,20156580,

%U 20030039,78152535,84672780,303906051,354822776,1184959314,1476390160

%N Number of walks of length n between two adjacent nodes in the cycle graph C_9.

%C In general 2^n/m*Sum(r,0,m-1,Cos(2Pi*k*r/m)Cos(2Pi*r/m)^n) is the number of walks of length n between two nodes at distance k in the cycle graph C_m. Here we have m=9 and k=1.

%C Also, with offset 2, the cogrowth sequence of the 18-element group D9 = <S,T | S^9, T^2, (ST)^2>. - _Sean A. Irvine_, Nov 14 2024

%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(r=0..8, cos(2*Pi*r/9)^(n+1)).

%F G.f.: x(-1+x+2x^2-x^3)/((1+x)(-1+2x)(1-3x^2+x^3)).

%F a(n) = a(n-1) + 5*a(n-2) - 4*a(n-3) - 5*a(n-4) + 2*a(n-5).

%o (PARI) a(n) = round(2^n/9*sum(r=0, 8, cos(2*Pi*r/9)^(n+1))) \\ _Michel Marcus_, Jul 18 2013

%o (PARI) Vec( x*(-1+x+2*x^2-x^3)/((1+x)*(-1+2*x)*(1-3*x^2+x^3))+O(x^66) ) \\ _Joerg Arndt_, Jul 18 2013

%Y Cf. A095367, A095368, A095369.

%Y Cf. A007582 (D8), A377573 (D7).

%K nonn

%O 1,3

%A _Herbert Kociemba_, Jul 03 2004