AIME 2015 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 \(1, 2, \ldots, 7\).


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