Question 195
Let $n \ge 3$ be an integer. Determine, as a function of $n$, the number of circular arrangements $x_1, x_2, \dots, x_n$ of the numbers $1, 2, \dots, n$ such that
$$
\sum_{i=1}^{n} |x_i - x_{i+2}| = 2n - 4,
$$
where the indices $i$ and $i+2$ are to be interpreted modulo $n$. Note that any rotation of a circular arrangement is considered to be the same circular arrangement, i.e., the circular arrangements $x_1, x_2, x_3, \dots, x_n$ and $x_2, x_3, \dots, x_n, x_1$ are considered to be the same.
$$
\sum_{i=1}^{n} |x_i - x_{i+2}| = 2n - 4,
$$
where the indices $i$ and $i+2$ are to be interpreted modulo $n$. Note that any rotation of a circular arrangement is considered to be the same circular arrangement, i.e., the circular arrangements $x_1, x_2, x_3, \dots, x_n$ and $x_2, x_3, \dots, x_n, x_1$ are considered to be the same.