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

New formulae are presented for the number $P(b)$ of non-negative integer solutions of a Diophantine equation ${\sum }_{i=1}^n{a}_i{x}_i=b$ and for the number $Q(b)$ of non-negative integer solutions of the Diophantine inequality ${\sum }_{i=1}^n{a}_i{x}_i\le b$ $(a_i>0,b\ge 0$).