2015 AIME II Problem 10

Call a permutation a_{1}, a_{2}, \ldots, a_{n} of the integers 1,2, \ldots, n quasi-increasing if a_{k} \leq a_{k+1}+2 for each 1 \leq k \leq n-1. For example, 53421 and 14253 are quasi-increasing permutations of the integers 1,2,3,4,5, but 45123 is not. Find the number of quasi-increasing permutations of the integers 1,2, \ldots, 7.