TITLE: Number of monomials in three variables of quasihomogeneus degree d
QUESTION [5 upvotes]: 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 $\alpha$.

REPLY [5 votes]: There is an asymptotic formula: If $w_0,...,w_r$ are nonnegative integers such that $gcd(w_0,...,w_r)=1$, then for large $n$: 

$$a_n := |\lbrace (k_0,...,k_r) \in \mathbb{N}_0^r \mid \sum_i k_iw_i = n \rbrace| = \frac{n^r}{r!\;w_0 \cdots w_r} + O(n^{r-1})$$

For, as in Francesco's answer we have 
$$\sum_{n \ge 0} a_n t^n = \frac{1}{(1-t^{w_0})\ldots (1-t^{w_r})}$$
(let me know if you need more details for this step). Now the result follows from Robert Israel's answer of the following question: 
Asymptotics for the coefficients of a rational function 

Moreover, it's known that for large $n$, $a_n$ is a "polynomial with periodic coefficients" in $n$, i.e. there are polynomials $f_i$ over $\mathbb{Q}$ of degree $r$ such that $a_{nw+i} = f_i(n)$ for $0 \le i < w$ and all large enough $n$, where $w := lcm(w_0,...,w_r)$. 
This is just Excercise 10.12 in [Eisenbud: Commutative Algebra].