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%I #18 Jul 16 2020 06:28:03
%S 2,14,72,330,1430,6008,24786,101118,409640,1652090,6643782,26667864,
%T 106914242,428292590,1714834440,6863694378,27466183286,109894593848,
%U 439656551730,1758830875230,7035859329512,28144840135514
%N Number of walks of length 2n+1 between two nodes at distance 5 in the cycle graph C_10.
%C 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
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-13,4).
%F a(n) = 4^n/5*Sum_{r=0..9} (-1)^r*Cos(Pi*r/5)^(2n+1).
%F a(n) = 7a(n-1)-13a(n-2)+4a(n-3).
%F G.f.: -2x^2/((-1+4x)(1-3x+x^2)).
%F 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
%p 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
%t 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}]
%K nonn
%O 2,1
%A _Herbert Kociemba_, Jul 12 2004
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