Question 39
Find all periodic sequences $a_1 a_2, \dots$ of real numbers such that the following conditions hold for all $n \ge 1$:
$$
a_{n+2} + a_n^2 = a_n + a_{n+1}^2 \quad \text{and} \quad |a_{n+1} - a_n| \le 1.
$$
$$
a_{n+2} + a_n^2 = a_n + a_{n+1}^2 \quad \text{and} \quad |a_{n+1} - a_n| \le 1.
$$