Hyper Binary Representation and Generating function

Question

Consider the recurrence relation: $f(0)=1$, for $n\geq1$, $f(2n+1) = f(n)$, $f(2n) = f(n) + f(n-1)$.

This sequence is known to represent the number of different ways $n$ can be expressed as a sum of integer powers of $2$ using each power no more than twice. The generating function can be written as $$\prod_ {i=0}^{\infty} (1+x^{2^i}+x^{2^{i+1}})$$ Is there a way to use this generating function to explain the hyper binary representation?

Answer 1

Let $F(x)$ be the hyper binary generating function. Then $$F(x)=(1+x+x^2)F(x^2)$$ and the recurrence relationship comes from looking at the coefficients.