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A166168 G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).

%I A166168 #11 Dec 19 2018 03:31:03
%S A166168 1,1,4,29,585,34212,5600397,2490542953,2968152042068,9416588994339205,
%T A166168 79216509536543420965,1762508872870620792746360,
%U A166168 103525263562786817866762466405,16031370626878431551103688398524485
%N A166168 G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).
%C A166168 Conjectured to consist entirely of integers.
%C A166168 The Lucas numbers (A000204) forms the logarithmic derivative of the Fibonacci numbers (A000045).
%C A166168 Note that Lucas(n^2) = [(1+sqrt(5))/2]^(n^2) + [(1-sqrt(5))/2]^(n^2).
%H A166168 G. C. Greubel, <a href="/A166168/b166168.txt">Table of n, a(n) for n = 0..60</a>
%H A166168 Sawian Jaidee, Patrick Moss, Tom Ward, <a href="https://arxiv.org/abs/1809.09199">Time-changes preserving zeta functions</a>, arXiv:1809.09199 [math.DS], 2018.
%F A166168 a(n) = (1/n)*Sum_{k=1..n} Lucas(k^2)*a(n-k), a(0)=1.
%F A166168 Logarithmic derivative yields A166169.
%e A166168 G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 585*x^4 + 34212*x^5 +...
%e A166168 log(A(x)) = x + 7*x^2/2 + 76*x^3/3 + 2207*x^4/4 + 167761*x^5/5 + 33385282*x^6/6 +...+ Lucas(n^2)*x^n/n +...
%p A166168 with(combinat): seq(coeff(series(exp(add((fibonacci(k^2-1)+fibonacci(k^2+1))*x^k/k,k=1..n)),x,n+1), x, n), n = 0 .. 15); # _Muniru A Asiru_, Dec 18 2018
%t A166168 CoefficientList[Series[Exp[Sum[LucasL[n^2]*x^n/n, {n, 1, 200}]], {x, 0, 50}], x](* _G. C. Greubel_, May 06 2016 *)
%o A166168 (PARI) {a(n)=polcoeff(exp(sum(m=1,n,(fibonacci(m^2-1)+fibonacci(m^2+1))*x^m/m)+x*O(x^n)),n)}
%Y A166168 Cf. A166169, A156216, A155200, A000204, A000045.
%K A166168 nonn
%O A166168 0,3
%A A166168 _Paul D. Hanna_, Oct 08 2009