---
id: 5900f41c1000cf542c50ff2e
challengeType: 5
title: 'Problem 175: Fractions involving the number of different ways a number can be expressed as a sum of powers of 2'
---
## Description
Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice.
For example, f(10)=5 since there are five different ways to express 10:10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1
It can be shown that for every fraction p/q (p>0, q>0) there exists at least one integer n such that f(n)/f(n-1)=p/q.
For instance, the smallest n for which f(n)/f(n-1)=13/17 is 241.
The binary expansion of 241 is 11110001.
Reading this binary number from the most significant bit to the least significant bit there are 4 one's, 3 zeroes and 1 one. We shall call the string 4,3,1 the Shortened Binary Expansion of 241.
Find the Shortened Binary Expansion of the smallest n for which f(n)/f(n-1)=123456789/987654321.
Give your answer as comma separated integers, without any whitespaces.
## Instructions
## Tests
```yml
tests:
- text: 'euler175() should return 1, 13717420, 8.'
testString: 'assert.strictEqual(euler175(), 1, 13717420, 8, "euler175() should return 1, 13717420, 8.");'
```
## Challenge Seed
```js
function euler175() {
// Good luck!
return true;
}
euler175();
```