We say a permutation `\sigma=\sigma_1\sigma_2\cdots\sigma_n` of length of `n` avoids `123` if there does not exist integers `i \lt j \lt k` such that `\sigma_i \lt \sigma_j \lt \sigma_k`.
`a(n)` is the number of permutations of length `n` avoiding `123`. For example, `a(3)=5` and `a(4)=14` .
Find `a(n)`.