Sums of Powers of Two
Define $f(0)=1$ and $f(n)$ to be the number of different ways $n$ can be expressed as a sum of integer powers of $2$ using each power no more than twice.
For example, $f(10)=5$ since there are five different ways to express $10$:
\begin{align} & 1 + 1 + 8\\ & 1 + 1 + 4 + 4\\ & 1 + 1 + 2 + 2 + 4\\ & 2 + 4 + 4\\ & 2 + 8 \end{align}What is $f(10^{25})$?