OFFSET
1,2
COMMENTS
In general 2^n/m*Sum_{r=0..m-1} 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=3.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-13,4).
FORMULA
a(n) = 4^n/5*Sum_{r=0..9} cos(3*Pi*r/5)*cos(Pi*r/5)^(2*n+1).
a(n) = 7*a(n-1)-13*a(n-2)+4*a(n-3).
G.f.: (-x+2*x^2)/((-1+4*x)*(1-3*x+x^2)).
a(n) = (2^(1+2*n)-(1/2*(3-sqrt(5)))^n-(1/2*(3+sqrt(5)))^n)/5. - Colin Barker, Apr 27 2016
E.g.f.: (2*exp(4*x) - exp(((3 - sqrt(5))*x)/2) - exp(((3 + sqrt(5))*x)/2))/5. - Ilya Gutkovskiy, Apr 27 2016
From Greg Dresden, Jan 19 2023: (Start)
a(n) = Sum_{k>0} binomial(2*n,n+k)-binomial(2*n,n+5k).
5*a(n) = 2*4^n - Lucas(2*n). (End)
MATHEMATICA
f[n_]:=FullSimplify[TrigToExp[(4^n/5)Sum[Cos[3Pi*k/5]Cos[Pi*k/5]^(2n+1), {k, 0, 9}]]]; Table[f[n], {n, 1, 35}]
PROG
(PARI) Vec((-x+2*x^2)/((-1+4*x)*(1-3*x+x^2)) + O(x^50)) \\ Colin Barker, Apr 27 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Herbert Kociemba, Jul 12 2004
STATUS
approved