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

1, 1, 4, 29, 585, 34212, 5600397, 2490542953, 2968152042068, 9416588994339205, 79216509536543420965, 1762508872870620792746360, 103525263562786817866762466405, 16031370626878431551103688398524485

Offset: 0

Author

Paul D. Hanna, Oct 08 2009

Keywords

• nonn

Comments

Conjectured to consist entirely of integers.

The Lucas numbers (A000204) forms the logarithmic derivative of the Fibonacci numbers (A000045).

Note that Lucas(n^2) = [(1+sqrt(5))/2]^(n^2) + [(1-sqrt(5))/2]^(n^2).

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 585*x^4 + 34212*x^5 +...
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 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..60
• Sawian Jaidee, Patrick Moss, Tom Ward, Time-changes preserving zeta functions, arXiv:1809.09199 [math.DS], 2018.

Crossrefs

Cf. A166169, A156216, A155200, A000204, A000045.

Programs

Maple

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

Mathematica

CoefficientList[Series[Exp[Sum[LucasL[n^2]*x^n/n, {n, 1, 200}]], {x, 0, 50}], x](* G. C. Greubel, May 06 2016 *)

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)}

Formula

a(n) = (1/n)*Sum_{k=1..n} Lucas(k^2)*a(n-k), a(0)=1.

Logarithmic derivative yields A166169.