Conservative Sequences

A finite sequence $\{a_n\}$ with $k \geqslant 2$ terms is called conservative if

Let $T_m$ be the number of permutations of $1, 2, \ldots, m$ that form conservative sequences, and $S_m$ the number of such permutations that moreover start with $1$. By convention $S_1 = T_1 = 1$.

1. Find $S_4$ and $T_4$. 2. Find a closed formula for the sequence $\{S_{n+3} - S_{n+2} - S_n\}$. 3. Find a closed formula for the sequence $\{T_{n+3} - T_{n+2} - T_n\}$.