OIM 2018 Problem 5

Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define $$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$ We say that the permutation $a_1, a_2, \dots, a_n$ is guadiana if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?