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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 $r\in \mathbb{N}$ ) :

$x_1+x_2=c_1$

$x_3+x_4=c_2$

...

$x_{2n-1}+x_{2n}=c_n$

$|x_1|+|x_2|+\ldots+|x_{2n}|\leq r$

$x_1,\ldots,x_{2n} \in \mathbb{Z}$

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    Note that the proposed duplicate uses an *equality constraint*, – 2017-01-23
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    The duplicate asks about finding all solutions but here we want to find the number of solutions (or an upper bound) without actually calculating them. – 2017-01-29

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