A203858 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^7, where Lucas(n) = A000204(n).

1, 1093, 5461, 205339, 3897434, 102033577, 2464268044, 63327787115, 1627243839488, 42592757748839, 1123514934469218, 29909548355299575, 801531714260597080, 21610508530971123358, 585611144766061271010, 15940294818101008311105, 435592135387553867410170

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

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)^8 * x^n/n), which is the g.f. of A203808.

a(n) ~ phi^(7*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)^1093 * (1-4*x^3-x^6)^5461 * (1-7*x^4+x^8)^205339 * (1-11*x^5-x^10)^3897434 * (1-18*x^6+x^12)^102033577 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^8 * x^n/n ) = g.f. of A203808:
F(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
where
log(F(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#]*LucasL[#]^7 &]/n; Array[a, 50] (* G. C. Greubel, Mar 05 2018 *)

Prog

(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^7)/n)}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^8*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. A203808, A203853, A203854, A203855, A203856, A203857, A203859, A203800.

Sequence in context: A270833 A273471 A266829 * A115192 A307220 A091674

Adjacent sequences: A203855 A203856 A203857 * A203859 A203860 A203861

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2012

Status

approved