Number of solutions of linear equation in $n$ unknowns with box constraints

Question

Given the following linear equation in $x_1, x_2, \dots, x_n \in \mathbb positive\ integers\ including\ 0 $

$$C_1 x_1 + C_2 x_2 + C_3 x_3 + \dots + C_n x_n = 0$$

where $C_1, C_2, \dots, C_n \in \mathbb Z$ $$L \leq x_1,x_2,\dots,x_n \leq R$$

L and R are positive integers including 0

how can I find the number of solutions?

I'm confused: both $C_i$ and $x_i$ are non negative? Sum of non negative numbers is non negative making it rather trivial. — Radost, Oct 07 '19 at 21:15
@RodrigodeAzevedo $\mathbb{N}$ is ambiguous. See https://math.stackexchange.com/questions/283/is-0-a-natural-number — Radost, Oct 07 '19 at 21:19
@Radost sorry for the mistake $C_i$s belong to integer set both positive and negative — Pritish Nayak, Oct 09 '19 at 18:57

score 0 · Answer 1 · answered Oct 09 '19 at 19:57

Recursive approach leads to solution with $\mathcal{O}(n \max(|L|,|R|))$ by reducing this to problem of selecting sums from multiset $\{C_1,C_1,\dots,C_n\}$ with appropriate number of repetitions.

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Wiki: Subset sum problem

Number of solutions of linear equation in $n$ unknowns with box constraints

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