A203853 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^2, where Lucas(n) = A000204(n).

1, 4, 5, 10, 24, 50, 120, 270, 640, 1500, 3600, 8610, 20880, 50700, 124024, 304290, 750120, 1854400, 4600200, 11440548, 28527320, 71289000, 178526880, 447910470, 1125750120, 2833885800, 7144449920, 18036373140, 45591631800, 115381697740, 292329067800, 741410800830

Offset

1,2

Comments

Apparently the same as A032170, if n > 2. - R. J. Mathar, Jan 11 2012

Links

Paul D. Hanna, Table of n, a(n) for n = 1..640

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)^3 * x^n/n), which is the g.f. of A203803.

a(n) ~ phi^(2*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)^4 * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^10 * (1-11*x^5-x^10)^24 * (1-18*x^6+x^12)^50 * (1-29*x^7-x^14)^120 * ... * (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) * ...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^3 * x^n/n ) = g.f. of A203803:
F(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 + ...
where
log(F(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 + ... + Lucas(n)^3*x^n/n + ...

Mathematica

a[n_]:= 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^2 &]; Array[a, 30] (* G. C. Greubel, Dec 25 2017 *)

Prog

(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^2)/n)}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^3*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. A001254, A203803, A006206, A203854, A203855, A203856, A203857, A203858, A203859, A203800.

Sequence in context: A166577 A242960 A288141 * A109675 A052508 A305887

Adjacent sequences: A203850 A203851 A203852 * A203854 A203855 A203856

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2012

Status

approved