A203319 a(n) = n*fibonacci(n) * Sum_{d|n} 1/(d*fibonacci(d)).

1, 3, 7, 19, 26, 81, 92, 267, 358, 848, 980, 3061, 3030, 7976, 11042, 25099, 27150, 78642, 79440, 219884, 270704, 584862, 659112, 1977909, 1950651, 4735370, 6204499, 14189096, 14912642, 43168586, 41734340, 110786987, 135815060, 290854380, 339428752, 953889058, 893839230

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

Equals the logarithmic derivative of A203318.

L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:

(1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} F(n*k+n)/F(n) * x^(n*k) where F(n) = fibonacci(n).

(2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) ) where Lucas(n) = A000032(n).

(3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) ) where Lucas(n) = A000032(n).

(4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.

Example

L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
where
L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
here F(n) = fibonacci(n) = A000045(n).
Equivalently,
L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
here Lucas(n) = A000032(n).
Exponentiation of the l.g.f. equals the g.f. of A203318:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...

Mathematica

a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* Jean-François Alcover, Dec 23 2015 *)

Prog

(PARI) {a(n)=if(n<1, 0, n*fibonacci(n)*sumdiv(n, d, 1/(d*fibonacci(d))) )}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1, n+1, (x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))), n)}
(PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L, n)}
(PARI) {a(n)=local(A=1+x+x*O(x^n), F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

Crossrefs

Cf. A203318, A203321; A203414 (Pell variant).

Cf. A000032 (Lucas), A000045 (Fibonacci), A001906, A001076, A004187, A049666, A049660, A049667.

Sequence in context: A226923 A001985 A203321 * A366171 A164097 A171140

Adjacent sequences: A203316 A203317 A203318 * A203320 A203321 A203322

Keyword

nonn

Author

Paul D. Hanna, Jan 01 2012

Status

approved