On the number of solutions of the Diophantine equation of Frobenius

We consider a linear Diophantine equation of the form
$$ x_1 a_1+\ldots+x_n a_n = N, $$
where $n$ is a fixed integer greater than one, $0<a_1<\ldots<a_n$ is a fixed set of integers such that $(a_1,\ldots,a_n)=1$. We denote by $f(N)$ the number of solutions in non-negative integers. It is well known that $f(x)=P(x)+\Delta(x)$, where $P(x)$ is a polynomial in $x$ of degree $n-1$ and $\Delta(x)$ is a periodic function with period $a_1\ldots a_n$. We apply an elementary approach to the problem of calculating $\Delta(x)$, and utilize roots of unity arguments in constructing this periodic function. For $f(N)$, an explicit expression is obtained for arbitrary $n$; this expression includes complicated sums containing the roots of unity. In the case $n=2$, this approach leads to a computable explicit expression for $f(x)$. We note that previously the expression for $\Delta(x)$ has not been known.