A095933 Number of walks of length 2n+1 between two nodes at distance 5 in the cycle graph C_10.

2, 14, 72, 330, 1430, 6008, 24786, 101118, 409640, 1652090, 6643782, 26667864, 106914242, 428292590, 1714834440, 6863694378, 27466183286, 109894593848, 439656551730, 1758830875230, 7035859329512, 28144840135514

Offset

2,1

Comments

In general 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=10 and k=5. Herbert

Links

Table of n, a(n) for n=2..23.

Index entries for linear recurrences with constant coefficients, signature (7,-13,4).

Formula

a(n) = 4^n/5*Sum_{r=0..9} (-1)^r*Cos(Pi*r/5)^(2n+1).

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

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

a(n) = (8/5)*4^n+2/5*(sqrt(5)-2)*2^n*(3+sqrt(5))^(-n)-2/5*(sqrt(5)+2)*2^n*(3-sqrt(5))^(-n). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 24 2008

Maple

f:= gfun:-rectoproc({- a(n) + 7*a(n-1) - 13*a(n-2) + 4*a(n-3), a(2)=2, a(3)=14, a(4)=72}, a(n), remember): map(f, [$2..23]); # Georg Fischer, Jul 16 2020

Mathematica

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

Crossrefs

Sequence in context: A171012 A094583 A002058 * A263218 A189305 A043011

Adjacent sequences: A095930 A095931 A095932 * A095934 A095935 A095936

Keyword

nonn

Author

Herbert Kociemba, Jul 12 2004

Status

approved