A095307 Number of walks of length n between two nodes at distance 2 in the cycle graph C_7.

1, 0, 4, 1, 15, 7, 56, 37, 210, 176, 793, 793, 3017, 3458, 11561, 14756, 44592, 62017, 172995, 257775, 674520, 1062601, 2641366, 4352660, 10381281, 17742621, 40927033, 72048354, 161766061, 291693136, 640758252, 1178135905, 2542557383, 4749439975, 10103745288

Offset: 2

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=7 and k=2.

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-2).

Programs

Mathematica

LinearRecurrence[{1,4,-3,-2},{1,0,4,1},40] (* Harvey P. Dale, Sep 22 2019 *)

PARI

Vec(x^2*(1-x)/((1-2*x)*(1+x-2*x^2-x^3)) + O(x^40)) \\ Colin Barker, Nov 28 2015

Formula

a(n) = (2^n/7)*Sum_{r=0..6} cos(4*Pi*r/7)*cos(2*Pi*r/7)^n.

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

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