Let $k, l$ be positive integers and $k_1, l_1$ non-negative integers. I have a polyhedral give by the following inequalities: $$ A = \{(r_1,r_3,r_4,r_5): r_1+3r_4\leq l_1, -k_1 \leq r_5-r_3 \leq 0, r_4 >0, r_5 - r_3 + r_4 \leq k-k_1 \}. $$ Are there some method to compute the number of integer points in the polyhedral $A$? The number of integers in $A$ is some function $a_{k,l,k_1,l_1}$ in $k,l,k_1,l_1$. Thank you very much.
Edit: maybe we need to use Ehrhart polynomial?