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In 2000, A. Tripathi used generating functions to obtain a formula for thenumber of non-negative solutions (x,y) of the equation ax + by = n where a, band n are given positive integers. We generalize this procedure for the numberof solutions of the equation ax + by + cz = n. The formula leads us to asurprising 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 toprove the equivalence between two fundamental results by Gauss and Sylvester inthe nineteenth century which are generally viewed to be independent.

Number of Non-negative Integer Solutions of the Equation ax+by+cz=n and its Relation with Quadratic Residues