I want to find the number of integer solutions for the following system of linear equations without computing the actual solutions ($c_1,\ldots,c_n \in \mathbb{Z}$ and $r\in \mathbb{N}$ ) :
$x_1+x_2=c_1$
$x_3+x_4=c_2$
...
$x_{2n-1}+x_{2n}=c_n$
$|x_1|+|x_2|+\ldots+|x_{2n}|\leq r$
$x_1,\ldots,x_{2n} \in \mathbb{Z}$