OBM 1991 Problem 3

Given $k > 0$ , the sequence $a_n$ is defined by its first two members and $$a_{n+2} = a_{n+1} + \frac{k}{n}a_n$$ a)For which $k$ can we write $a_n$ as a polynomial in $n$ ?

b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$ ? ( $p,q$ are polynomials in $\mathbb R[X]$ ).