Question 86
Let $n \ge 3$ be an integer and let $(p_1, p_2, p_3, \dots, p_n)$ be a permutation of $\{1, 2, 3, \dots, n\}$. For this permutation we say that $p_t$ is a *turning point* if $2 \le t \le n-1$ and
$$
(p_t - p_{t-1})(p_t - p_{t+1}) \gt 0.
$$
For example, for $n=8$, the permutation $(2, 4, 6, 7, 5, 1, 3, 8)$ has two turning points: $p_4 = 7$ and $p_6 = 1$.
For fixed $n$, let $q(n)$ denote the number of permutations of $\{1, 2, 3, \dots, n\}$ with exactly one turning point.
Find all $n \ge 3$ for which $q(n)$ is a perfect square.
$$
(p_t - p_{t-1})(p_t - p_{t+1}) \gt 0.
$$
For example, for $n=8$, the permutation $(2, 4, 6, 7, 5, 1, 3, 8)$ has two turning points: $p_4 = 7$ and $p_6 = 1$.
For fixed $n$, let $q(n)$ denote the number of permutations of $\{1, 2, 3, \dots, n\}$ with exactly one turning point.
Find all $n \ge 3$ for which $q(n)$ is a perfect square.