## Exploring the number of different ways a number can be expressed as a sum of powers of 2

### Problem 169

Published on Friday, 23rd November 2007, 09:00 pm; Solved by 3022; Difficulty rating: 50%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:

1 + 1 + 8

1 + 1 + 4 + 4

1 + 1 + 2 + 2 + 4

2 + 4 + 4

2 + 8

What is f(10^{25})?