Exercise 7

Consider the inequality $$x_1+x_2+x_3+x_4\leq n$$ where $x_1,x_2,x_3,x_4,n\geq 0$ are all integers. Suppose also that $x_2\geq 2$, $x_3$ is a multiple of $4$, and $0\leq x_4\leq 3$. Let $c_n$ be the number of solutions of the inequality subject to these restrictions. Find the generating function for the sequence $\{c_n\colon n\geq 0\}$ and use it to find a closed formula for $c_n$.