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

%I A187780 #15 Jul 19 2016 11:31:10
%S A187780 1,3,6,13,31,84,271,1111,6096,44965,434321,5388944,85434621,
%T A187780 1727597731,44466614106,1455616862597,60619117448531,3211943842710212,
%U A187780 216483614502128251,18558646821817827015,2023790814160269113876,280732940929438329958733,49535201863823517417076181
%N A187780 Sum_{k=0..n} Lucas(k)^(n-k).
%H A187780 Vincenzo Librandi, <a href="/A187780/b187780.txt">Table of n, a(n) for n = 0..100</a>
%F A187780 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 A187780 G.f.: A(x) = Sum_{n>=0} x^n/(1 - Lucas(n)*x).
%t A187780 Table[Sum[LucasL[k]^(n-k), {k, 0, n}], {n, 0, 20}]
%t A187780 (* constants: *)
%t A187780 ceven = N[Sum[((1+Sqrt[5])/2)^(-k^2), {k, -Infinity, +Infinity}], 50]
%t A187780 codd = N[Sum[((1+Sqrt[5])/2)^(-(k+1/2)^2), {k, -Infinity, +Infinity}], 50]
%o A187780 (PARI) Lucas(n)=fibonacci(n-1)+fibonacci(n+1)
%o A187780 a(n)=sum(k=0, n, Lucas(k)^(n-k))
%o A187780 for(n=0,21,print1(a(n),", ")) \\ _Paul D. Hanna_, Jan 05 2013
%Y A187780 Cf. A000032, A135961.
%K A187780 nonn
%O A187780 0,2
%A A187780 _Vaclav Kotesovec_, Jan 05 2013