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 $p / q$ ($p \gt 0$, $q \gt 0$) 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.
solutions/s175.md.