A203854 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^3, where Lucas(n) = A000204(n).
1, 13, 21, 79, 266, 957, 3484, 12935, 48768, 185951, 716418, 2781675, 10878520, 42789478, 169181010, 671866245, 2678678730, 10716651456, 43007270292, 173072549610, 698235680844, 2823329210391, 11439823946306, 46440709210035, 188856966693230, 769241291729020
Offset: 1
Keywords
Examples
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) *...)
where F(x) = exp( Sum_{n>=1} Lucas(n)^4 * x^n/n ) = g.f. of A203804:
F(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
where
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 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..360
Programs
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Mathematica
a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^3&]; Array[a, 30] (* Jean-François Alcover, Dec 04 2015 *)
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PARI
{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^3)/n)} -
PARI
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)} {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))}
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)^4 * x^n/n), which is the g.f. of A203804.
a(n) ~ phi^(3*n) / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Sep 02 2017