## 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.

### To solve this problem, we will use the known properties of Stern’s Diatomic Series. Stern’s diatomic sequence (denoted F(n) in this problem) is defined by F(0) = 0, F(1) = 1 and for n > 1, F(n) = F(floor(n/2)) + F(ceil(n/2)).

First, observe a property of Stern’s sequence. If p and q are relatively prime positive integers and n is the smallest positive integer such that F(n)/F(n-1) = p/q. Then n reads in binary as 1_{e1} 0_{e2} 1_{e3}…0_{ek} 1_{ek+1} where the e’s are the partial quotients in the continued fraction for p/q.

The continued fraction of 123456789 / 987654321 can be calculated as [0; 8, 1, 3, 1, 4, 1, 1, 2, 2, 3, 9 ,4].

Using the property of the Stern’s sequence, we see that the Shortened Binary Expansion of the smallest n for which f(n) / f(n-1) = 123456789 / 987654321 is the sequence of numbers (excluding the first zero): 8, 1, 3, 1, 4, 1, 1, 2, 2, 3, 9, 4.

So the answer should be given as: 8,1,3,1,4,1,1,2,2,3,9,4. This would be the Shortened Binary Expansion of the required number.

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