Question 8
Given $k \gt 0$, the sequence $a_1, a_2, a_3, \dots$ is defined by its first two members and $a_{n+2} = a_{n+1} + \frac{k}{n} a_n$. For which $k$ can we write an $a_n$ as a polynomial in $n$? For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$, where $p$ and $q$ are polynomials?