Solution:
Answer: $\quad \frac{1}{6}(n-6)(n-1)(n+1)$

Consider the first point of the path that lies on $x=3$. There are two possibilities for this point: $(3,0)$ and $(3,1)$, and there is exactly one valid way of getting to each point from the origin. Similarly, consider the last point of the path that lies on $x=n-3$. There are two possibilities: $(n-3,2)$ and $(n-3,3)$, and there is exactly one valid way of getting to the destination from each of the two points. Now we count the number of valid paths from each of $(3,0)$ and $(3,1)$, to each of $(n-3,2)$ and $(n-3,3)$, and the answer will be the sum.

- From $(3,1)$ to $(n-3,2)$: there are no forbidden points along the way, so there are $n-5$ ways.
- From $(3,0)$ to $(n-3,2)$: the path must not pass through $(n-3,0)$, and there is exactly one invalid path. So there are $\binom{n-4}{2}-1$ ways.
- From $(3,1)$ to $(n-3,3)$: the path must not pass through $(3,3)$, and there is exactly one invalid path. So there are $\binom{n-4}{2}-1$ ways.
- From $(3,0)$ to $(n-3,3)$: the path must not pass through $(n-3,0)$ and $(3,3)$, and there are exactly two invalid paths. So there are $\binom{n-3}{3}-2$ ways.

Summing, we obtain the answer:
$$n-5+\binom{n-4}{2}-1+\binom{n-4}{2}-1+\binom{n-3}{3}-2=\frac{n^{3}-6 n^{2}-n+6}{6}=\frac{(n-6)(n-1)(n+1)}{6}.$$