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%I #13 Sep 02 2017 05:04:40
%S 1,13,21,79,266,957,3484,12935,48768,185951,716418,2781675,10878520,
%T 42789478,169181010,671866245,2678678730,10716651456,43007270292,
%U 173072549610,698235680844,2823329210391,11439823946306,46440709210035,188856966693230,769241291729020
%N a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^3, where Lucas(n) = A000204(n).
%H Paul D. Hanna, <a href="/A203854/b203854.txt">Table of n, a(n) for n = 1..360</a>
%F G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^4 * x^n/n), which is the g.f. of A203804.
%F a(n) ~ phi^(3*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Sep 02 2017
%e G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^13 * (1-4*x^3-x^6)^21 * (1-7*x^4+x^8)^79 * (1-11*x^5-x^10)^266 * (1-18*x^6+x^12)^957 *...* (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) *...)
%e where F(x) = exp( Sum_{n>=1} Lucas(n)^4 * x^n/n ) = g.f. of A203804:
%e F(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
%e where
%e log(F(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...
%t a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^3&]; Array[a, 30] (* _Jean-François Alcover_, Dec 04 2015 *)
%o (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^3)/n)}
%o (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o {a(n)=local(F=exp(sum(m=1, n, Lucas(m)^4*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}
%Y Cf. A203804, A203853, A203855, A203856, A203857, A203858, A203859, A203800.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jan 07 2012
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