Question 8
Let $k$ be a positive integer. A sequence of integers $\langle a_i \rangle_{i=1}^\infty$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \dots, a_{n+k}\}$. Determine, as a function of $k$, how many $k$-pop sequences there are.