Let $S\_n = \{0,1\}^n$. 

Let $S\_{i,n} \subseteq S\_n$ be a set such that $s \in S\_{i,n}$ if and only if $s$ contains exactly $i$ occurences of $11$.

Example for $n=3$.
\begin{align\*}
S\_3 &= \{000, 001, 010, 011, 100, 101, 110, 111\}\\
S\_{0,3} &= \{000,001,010,100,101\}\\
S\_{1,3} &= \{011,110\} \\
S\_{2,3} &=\{111\}
\end{align\*}

Example for $n=4$.
\begin{align\*}
S\_4 &= \{0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111,
1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111
\}\\
S\_{0,4} &= \{0000,0001,0010,0100,0101,1000,1001,1010\}\\
S\_{1,4} &= \{0110,0110,1100, 1011,1101\} \\
S\_{2,4} &=\{0111,1110\}\\
S\_{3,4} &=\{1111\}
\end{align\*}
Therefore $|S\_{0,4}| = 8 , |S\_{1,4}| = 5, |S\_{2,4}| = 2, |S\_{3,4}| = 1$. 

Is it possible to find a formula for $|S\_{i,n}|$ ? Upper and lower bounds would also be nice to have

This sequence can also be found in [OEIS](https://oeis.org/search?q=144%2C%20235%2C%20241%2C%20187%2C%20116%2C%2062%2C%2025%2C%2011%2C%202%2C%201&sort=&language=english&go=Search%22OEIS%22) but no formula is available