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A095933 Number of walks of length 2n+1 between two nodes at distance 5 in the cycle graph C_10. 0
2, 14, 72, 330, 1430, 6008, 24786, 101118, 409640, 1652090, 6643782, 26667864, 106914242, 428292590, 1714834440, 6863694378, 27466183286, 109894593848, 439656551730, 1758830875230, 7035859329512, 28144840135514 (list; graph; refs; listen; history; text; internal format)
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.

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)).

Recurrence: a(n)=7*a(n-1)-13*a(n-2)+4*a(n-3), where a(1)=2, a(2)=14, a(3)=72; formula 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

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

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Last modified February 21 03:06 EST 2019. Contains 320364 sequences. (Running on oeis4.)