A095364 Number of walks of length n between two adjacent nodes in the cycle graph C_9.

1, 0, 3, 0, 10, 0, 35, 1, 126, 11, 462, 78, 1716, 455, 6435, 2380, 24311, 11628, 92398, 54264, 352947, 245157, 1354102, 1081575, 5215250, 4686826, 20156580, 20030039, 78152535, 84672780, 303906051, 354822776, 1184959314, 1476390160

Offset: 1

Author

Herbert Kociemba, Jul 03 2004

Keywords

• nonn

Comments

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.

Also, with offset 2, the cogrowth sequence of the 18-element group D9 = . - Sean A. Irvine, Nov 14 2024

Links

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

Crossrefs

Cf. A095367, A095368, A095369.

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

Programs

PARI

a(n) = round(2^n/9*sum(r=0, 8, cos(2*Pi*r/9)^(n+1))) \\ Michel Marcus, Jul 18 2013

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

Formula

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

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

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