Putnam 2015 B5

Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that $$|i-j| = 1 \text{ implies } |\pi(i) -\pi(j)| \leq 2$$ for all $i,j$ in $\{1,2,\dots,n\}$. Show that for $n \geq 2$, the quantity $$P_{n+5} - P_{n+4} - P_{n+3} + P_n$$ does not depend on $n$, and find its value.