Example 4.9 Find the number of integer points (points with both coordinates as integers) within the triangle (including the boundary) formed by the lines $x+3y=12$, $x=0$, and $y=0$.

 untranslated text:
求由直线 $x+3 y=12$, 直线 $x=0$ 及直线 $y=0$ 所围成的三角形内(包括边界)的整点(横坐标和纵坐标均是整数的点)的个数. 

translated text:
Find the number of integer points (points with both coordinates as integers) within the triangle (including the boundary) formed by the lines $x+3y=12$, $x=0$, and $y=0$.
Solution: Let the required number be $N$, then $N$ is the number of non-negative integer solutions to the inequality $x+3 y \leqslant 12$. Let $z=12-x-3 y$, if $x+3 y \leqslant 12$, then $z \geqslant 0$ and $x+3 y+z=12$, so $N$ is the number of non-negative integer solutions to the equation $x+3 y+z=12$, hence $N$ is the coefficient of $t^{12}$ in the expansion of
$$
A(t)=\left(1+t+t^{2}+\cdots\right)^{2}\left(1+t^{3}+t^{6}+\cdots\right)
$$

and
$$
\begin{aligned}
A(t) & =(1-t)^{-2}\left(1-t^{3}\right)^{-1} \\
& =\left(1+t+t^{2}\right)^{2}\left(1-t^{3}\right)^{-3} \\
& =\left(1+2 t+3 t^{2}+2 t^{3}+t^{4}\right) \sum_{k=0}^{\infty}\binom{k+2}{2} t^{3 k},
\end{aligned}
$$

Therefore,
$$
\begin{aligned}
N & =\binom{4+2}{2}+2 \cdot\binom{3+2}{2} \\
& =15+20=35 .
\end{aligned}
$$