A203857 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^6, where Lucas(n) = A000204(n).

1, 364, 1365, 29230, 354312, 5667900, 84974760, 1347387210, 21411102720, 346282421940, 5645803690800, 92886793449030, 1538448587832240, 25635241395476100, 429333683845968552, 7222607529064709670, 121980435560782376760, 2067248664062116147200

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..795

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^7 * x^n/n), which is the g.f. of A203807.

a(n) ~ phi^(6*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017

Example

G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4)^364 * (1-4*x^3-x^6)^1365 * (1-7*x^4+x^8)^29230 * (1-11*x^5-x^10)^354312 * (1-18*x^6+x^12)^5667900 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^7 * x^n/n ) = g.f. of A203807:
F(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
where log(F(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^6&]/n; Array[a, 18] (* Jean-François Alcover, Dec 07 2015 *)

Prog

(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^6)/n)}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^7*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))}

Crossrefs

Cf. A203807, A203853, A203854, A203855, A203856, A203858, A203859, A203800.

Sequence in context: A131348 A303123 A043471 * A305184 A105920 A241617

Adjacent sequences: A203854 A203855 A203856 * A203858 A203859 A203860

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2012

Status

approved