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

### Problem 175

Published on Friday, 28th December 2007, 01:00 pm; Solved by 1164; Difficulty rating: 70%
Define f(0)=1 and f(

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

f(

For instance, the smallest

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

f(

Give your answer as comma separated integers, without any whitespaces.

`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 thatf(

`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 whichf(

`n`)/f(`n`-1)=123456789/987654321.Give your answer as comma separated integers, without any whitespaces.