We say permutation $\sigma=\sigma_1\sigma_2\sigma_3\cdots\sigma_n$ is an up-up-down permutation if $\sigma_1<\sigma_2<\sigma_3>\sigma_4<\sigma_5<\sigma_6>\sigma_7<\sigma_8<\sigma_9>\cdots$. For example, $\sigma=(1\ 5\ 6\ 2\ 4\ 7\ 3)$ is an up-up-down permutation of length $7$.
We say permutation $\sigma=\sigma_1\sigma_2\sigma_3\cdots\sigma_n$ is a derangement if $\sigma_k\ne k$ for any $1\leq k\leq n$.
$a(n)$ is the number of up-up-down derangements of length $n$. For example, $\sigma=(2\ 5\ 6\ 1\ 4\ 7\ 3)$ is an up-up-down derangement of length $7$. Number of up-down derangements is counted by sequence A129815 on OEIS.
Find $a(n)$.