A095369 Number of walks of length n between two nodes at distance 4 in the cycle graph C_9.

1, 1, 6, 7, 28, 36, 120, 165, 495, 716, 2003, 3018, 8024, 12512, 31977, 51357, 127110, 209475, 504736, 850840, 2003784, 3445885, 7956715, 13926276, 31609071, 56191734, 125640180, 226444616, 499685777, 911607609, 1988440598

Offset: 4

Author

Herbert Kociemba, Jul 03 2004

Keywords

• nonn
• easy

Comments

In general, (2^n/m)*Sum_{r=0..m-1} cos(2*Pi*k*r/m)*cos(2*Pi*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=4.

Links

• Michael De Vlieger, Table of n, a(n) for n = 4..3325
• Index entries for linear recurrences with constant coefficients, signature (1,5,-4,-5,2).

Crossrefs

Cf. A095364, A095367, A095368.

Programs

Mathematica

Drop[CoefficientList[Series[-x^4/((1 + x) (-1 + 2 x) (1 - 3 x^2 + x^3)), {x, 0, 34}], x], 4] (* Michael De Vlieger, Jan 23 2022 *)

Formula

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

G.f.: x^4/((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).