Question 68
A sequence of integers $a_0, a_1, a_2, \dots$ is called *kawaii* if $a_0 = 0$, $a_1 = 1$, and, for any positive integer $n$, we have
$$
(a_{n+1} - 3a_n + 2a_{n-1})(a_{n+1} - 4a_n + 3a_{n-1}) = 0.
$$
An integer is called *kawaii* if it belongs to a kawaii sequence.
Suppose that two consecutive positive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that 3 divides $m$, and that $m/3$ is kawaii.
$$
(a_{n+1} - 3a_n + 2a_{n-1})(a_{n+1} - 4a_n + 3a_{n-1}) = 0.
$$
An integer is called *kawaii* if it belongs to a kawaii sequence.
Suppose that two consecutive positive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that 3 divides $m$, and that $m/3$ is kawaii.