A095930 Number of walks of length 2n between two nodes at distance 2 in the cycle graph C_10.

1, 4, 15, 57, 220, 859, 3381, 13380, 53143, 211585, 843756, 3368259, 13455325, 53774932, 214978335, 859595529, 3437550076, 13748021995, 54986385093, 219930610020, 879683351911, 3518631073489, 14074256379660, 56296324109907, 225183460127725, 900729032983924

Offset: 1

Author

Herbert Kociemba, Jul 12 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=10 and k=2.

Equals INVERT transform of A014138: (1, 3, 8, 22, 64, 196, ...). - Gary W. Adamson, May 15 2009

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-13,4)

Crossrefs

Cf. A014138.

Programs

Mathematica

f[n_]:=FullSimplify[TrigToExp[(4^n/10)Sum[Cos[2Pi*k/5]Cos[Pi*k/5]^(2n), {k, 0, 9}]]];Table[f[n], {n, 1, 35}]

PARI

Vec((-x+3*x^2)/((-1+4*x)*(1-3*x+x^2)) + O(x^50)) \\ Colin Barker, Apr 27 2016

Formula

a(n) = (4^n/10)*Sum_{r=0..9} cos(2*Pi*r/5)*cos(Pi*r/5)^(2*n).

a(n) = 7*a(n-1) - 13*a(n-2) + 4*a(n-3).

G.f.: (-x+3*x^2)/((-1+4*x)*(1-3*x+x^2)).

a(n) = (4^n + Lucas(2*n-1))/5. With a(0) = 0, binomial transform of A098703. - Ross La Haye, May 31 2006

a(n) = (2^(-1-n)*(2^(1+3*n) - (3-sqrt(5))^n*(1+sqrt(5)) + (-1+sqrt(5))*(3+sqrt(5))^n))/5. - Colin Barker, Apr 27 2016

E.g.f.: (2*exp(4*x) + (-1 - sqrt(5))*exp(((3 - sqrt(5))*x)/2) + (-1 + sqrt(5))*exp(((3 + sqrt(5))*x)/2))/10. - Ilya Gutkovskiy, Apr 27 2016