Let a sequence of polynomials $\{f_n\}_{n=0}^\infty$ in $\mathbb{Q}[x,y]$ be given in the following way:
$$f_0=1,$$
$$f_1=-x,$$
$$f_2=x^2-y,$$
$$f_{n+2}=-xf_{n+1}-yf_n.$$
For each $n\geq 0$, find the smallest positive integer $k(n)$ such that $x^{k(n)}\in (f_n,f_{n+1})$. Here $(f_n,f_{n+1})$ is the ideal generated by $f_n, f_{n+1}$.
How to solve this problem? Thanks for the help!