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Sums of Powers of Two

 Published on Friday, 23rd November 2007, 09:00 pm; Solved by 5441;
Difficulty rating: 50%

Problem 169

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 $f(10^{25})$?