OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4), where c = Sum_{k=-Infinity..Infinity} ((1+sqrt(5))/2)^(-k^2) = 2.555093469444518777230568... if n is even and c = Sum_{k=-Infinity..Infinity} ((1+sqrt(5))/2)^(-(k+1/2)^2) = 2.555093456793304790966994... if n is odd
G.f.: A(x) = Sum_{n>=0} x^n/(1 - Lucas(n)*x).
MATHEMATICA
Table[Sum[LucasL[k]^(n-k), {k, 0, n}], {n, 0, 20}]
(* constants: *)
ceven = N[Sum[((1+Sqrt[5])/2)^(-k^2), {k, -Infinity, +Infinity}], 50]
codd = N[Sum[((1+Sqrt[5])/2)^(-(k+1/2)^2), {k, -Infinity, +Infinity}], 50]
PROG
(PARI) Lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=sum(k=0, n, Lucas(k)^(n-k))
for(n=0, 21, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 05 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jan 05 2013
STATUS
approved