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 $10$:

$$\begin{array}{rl}& 1+1+8\\ & 1+1+4+4\\ & 1+1+2+2+4\\ & 2+4+4\\ & 2+8\end{array}$$

What is $f({10}^{25})$?