Denumerants and Their Approximations

Let \(a, b, c\) be fixed, pairwise relatively prime integers. We investigate the number of non-negative integral solutions of the equation \(ax + by + cz = n\) as a function of \(n\). We present a new algorithm that computes the “closed form” of this function. This algorithm is simple and its time performance is better than the performance of yet known algorithms. We also recall how to approximate the abovementioned function by a polynomial and we derive bounds on the “error” of this approximation for the case \(a = 1\).