Question 55
Define the sequence $(a_n)_{n \ge 0}$ by $a_0 = 0$, $a_1 = 1$, $a_2 = 2$, $a_3 = 6$, and
$$
a_{n+4} = 2a_{n+3} + a_{n+2} - 2a_{n+1} - a_n, \quad n \ge 0.
$$
Prove that $n^2$ divides $a_n$ for infinitely many positive integers $n$.
$$
a_{n+4} = 2a_{n+3} + a_{n+2} - 2a_{n+1} - a_n, \quad n \ge 0.
$$
Prove that $n^2$ divides $a_n$ for infinitely many positive integers $n$.