Invert the formula F(n)= (r^n - (1-r)^n)/sqrt(5) where r=(1+sqrt(5))/2 by the 
Lagrange Inversion Formula (LIF).  Let X=sqrt(5)*F(n) and s=1-r so X=r^n - s^n.
So X=sum from j=0 to infinity of ((ln(r))^j - (ln(s))^j)/j! * n^j.

Then n = X + sum of X^j * 
sum over all sequences (b(2),b(3),....,) of nonnegative integers of
(-1)^(sum of b(i) from i=2 to j) * 
( (sum of b(i) from i=2 to j) + j-1)!/(j!) * product from i=2 to j of 
(((ln(r))^i - (ln(s))^i)/i!)^b(i) / (b(i)!) )
such that sum of (i-1)*b(i) from i=2 to j equals j-1
and pray for convergence everywhere.

I got this form of the LIF from page 264 of G.P. Egorychev's book,
"Integral Representations of Combinatorial Sums"