Eccoci.

A composition of $n$ is a sequence of positive integers which sum to $n$. Such a sequence can be split into runs, where a run is a maximal contiguous subsequence of equal terms.

For example, $2,2,1,1,1,3,2,2$ is a composition of $14$ consisting of four runs:

$2, 2\quad 1, 1, 1\quad 3 \quad 2, 2$

Let $F(n)$ be the number of compositions of $n$ where every run has odd length.

For example, $F(5)=10$:

\begin{align*} & 5 &&4,1 && 3,2 &&2,3 &&2,1,2\\ &2,1,1,1 &&1,4 &&1,3,1 &&1,1,1,2 &&1,1,1,1,1 \end{align*}

Find $F(10^5)$. Give your answer modulo $1111124111$.