A095368 Number of walks of length n between two nodes at distance 3 in the cycle graph C_9.

1, 0, 5, 1, 21, 8, 84, 45, 330, 221, 1287, 1015, 5006, 4488, 19465, 19380, 75753, 82365, 295261, 346104, 1152944, 1442101, 4510830, 5969561, 17682795, 24582663, 69448446, 100804436, 273241161, 411921832, 1076832989

Offset: 3

Author

Herbert Kociemba, Jul 03 2004

Keywords

• nonn
• easy

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=3.

Links

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

Crossrefs

Cf. A095364, A095367, A095369.

Programs

Mathematica

LinearRecurrence[{1,5,-4,-5,2},{1,0,5,1,21},40] (* Harvey P. Dale, Dec 21 2024 *)

Formula

a(n) = (2^n/9)*Sum_{r=0..8} cos(2Pi*r/3)*cos(2Pi*r/9)^n.

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

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