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New formulae are presented for the number $P(b)$ of non-negative integersolutions of a Diophantine equation $\sum_{i=1}^{n}a_ix_i=b$ and for the number$Q(b)$ of non-negative integer solutions of the Diophantine inequality$\sum_{i=1}^{n}a_ix_i\leq b$ $(a_i>0, b\geq 0$).

Formulae for the number of non-negative integer solutions of linear Diophantine equation and inequality