  In 2000, A. Tripathi used generating functions to obtain a formula for the
number of non-negative solutions (x,y) of the equation ax + by = n where a, b
and n are given positive integers. We generalize this procedure for the number
of solutions of the equation ax + by + cz = n. The formula leads us to a
surprising connection between the number of solutions of the equation ax + by +
cz = n and quadratic residues. As a consequence of our work, we are able to
prove the equivalence between two fundamental results by Gauss and Sylvester in
the nineteenth century which are generally viewed to be independent.
