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

Abstract: New formulae are presented for the number $P(b)$ of non-negative integer solutions 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 $\sum_{i=1}^{n}a_ix_i\leq b$ $(a_i>0, b\geq 0$).