I am interested in computing the number of natural solutions $(x_0,y_0,z_0)$ of the equation $w_0x+w_1y+w_2z=d$, with $w_0,w_1,w_2$ and $d$ natural numbers and $\gcd (w_i,w_j)=1$ . Or equivalently, the number of monomials in three variables of quasihomogeneus degree d and weights $(w_0,w_1,w_2)$. Does anybody now any formula in terms of $w_0,w_1,w_2$ and $d$ or any reference in the literature?

With the previous notations I really need to compute $a_{\alpha+w_0w_1w_2}-a_{\alpha}$ being $\alpha< w_0w_1w_2$. I do not know if there are some kind of relations among the terms of this series...I want to look at the properties of these series because I need an explicit expresion in terms of the weights and $\alpha$.