  We deal with the problem to find the number $P(b)$ of integer non-negative
solutions of an equation $\sum_{i=1}^{n} a_i x_i=b$, where $a_1,a_2,...,a_n$
are natural numbers and $b$ is a non-negative integer. As different from the
traditional methods of investigation of the function $P(b)$, in our study we do
not employ the techniques of number series theory, but use in the main the
properties of the Kronecker function and the elements of combinatorics. The
formula is derived to express $P(b)$, for an integer non-negative $b$, via
$P(r)$, $P(r+M)$,..., $P(r+(s-1)M)$ when $s\neq 0$, where $s=\left[
n-\dfrac{\sum_{i=1}^{n}a_i+r}{M} \right]$ and takes quite small values in some
particular cases; $M$ is the least common multiple of the numbers
$a_1,a_2,\ldots,a_n$, and $r$ is the remainder of $b$ modulo $M$. Also, the
recurrent formulas are derived to calculate $P(b)$, for any non-negative
integer $b$, which, in particular, are used in finding $P(r)$, $P(r+M)$,...,
$P(r+(s-1)M)$. For the case where $s=0$ and $a_1$, $a_2$,..., $a_n$ are
coprime, the explicit formula $P(b)=\dfrac{M^{n-1}}{a_1a_2\ldots
a_n}C^{n-1}_{\left[\dfrac{b}{M}\right]+n-1}$ is given. To illustrate the
proposed method, examples of finding the function $P(b)$ for linear Diophantine
equations with $2,3,7$ and $n$ variables are given.
