What is the Hilbert series of $R/I$ for $I = (F,G)$ where $F,G$ is a regular sequence on $R = k[x,y]$ with $\deg F \leq \deg G?$
Definition: A sequence $F,G$ is regular on $R$ if $F$ is a nonzero divisor of $R$ and $G$ is a nonzero divisor of $R/(F)$.
We have the following exact sequences: $$0\to R(-\deg f)\stackrel{f\cdot}\to R\to R/(f)\to0$$ $$0\to R/(f)(-\deg g)\stackrel{g\cdot}\to R/(f)\to R/(f,g)\to0$$ From the first we get $H_{R/(f)}(t)=(1-t^{\deg f})H_R(t)$, while from the second we obtain $H_{R/(f,g)}(t)=(1-t^{\deg g})H_{R/(f)}(t)$.
