%I #15 Jul 19 2016 11:31:10

%S 1,3,6,13,31,84,271,1111,6096,44965,434321,5388944,85434621,

%T 1727597731,44466614106,1455616862597,60619117448531,3211943842710212,

%U 216483614502128251,18558646821817827015,2023790814160269113876,280732940929438329958733,49535201863823517417076181

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

%H Vincenzo Librandi, <a href="/A187780/b187780.txt">Table of n, a(n) for n = 0..100</a>

%F 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

%F G.f.: A(x) = Sum_{n>=0} x^n/(1 - Lucas(n)*x).

%t Table[Sum[LucasL[k]^(n-k), {k, 0, n}], {n, 0, 20}]

%t (* constants: *)

%t ceven = N[Sum[((1+Sqrt[5])/2)^(-k^2), {k, -Infinity, +Infinity}], 50]

%t codd = N[Sum[((1+Sqrt[5])/2)^(-(k+1/2)^2), {k, -Infinity, +Infinity}], 50]

%o (PARI) Lucas(n)=fibonacci(n-1)+fibonacci(n+1)

%o a(n)=sum(k=0, n, Lucas(k)^(n-k))

%o for(n=0,21,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 05 2013

%Y Cf. A000032, A135961.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jan 05 2013