A187780 Sum_{k=0..n} Lucas(k)^(n-k).

1, 3, 6, 13, 31, 84, 271, 1111, 6096, 44965, 434321, 5388944, 85434621, 1727597731, 44466614106, 1455616862597, 60619117448531, 3211943842710212, 216483614502128251, 18558646821817827015, 2023790814160269113876, 280732940929438329958733, 49535201863823517417076181

Offset: 0

Author

Vaclav Kotesovec, Jan 05 2013

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000032, A135961.

Programs

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]

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

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).