Periodic oscillations of coefficients of power series that satisfy functional equations

It is shown that the coefficients a_n of the power series f(z) = ∑^∞ _n=1 a_nzⁿ which satisfies the functional equation f(z)=z+f(z²+z³). display periodic oscillations; a_n ∼ ( øⁿ n) u(logn as n → ∞, where ø = (1 + 5 ^{1 2}) 2 and u(x) is a positive, nonconstant, continuous function which is periodic with period log(4 - ø). Similar results are obtained for a wide class of power series that satisfy similar functional equations. Power series of these types are of interest in combinatorics and computer science since they often represent generating functions. For example, the nth coefficient of the power series satisfying (*) enumerates 2, 3-trees with n leaves.