Number of Components of Pretzel links

Question

I am interested in the number of components (denoted by $N$) of the Pretzel Link $(p_1,\ldots,p_n)$. I understand the following facts about this:

1. Each $p_i$ can be considered as $0$ or $1$, depends on the parity of $p_i$;
2. When $n$ and $p_i$'s are all odd, or when exactly one $p_i$ is even, $N=1$;
3. When $n$ is even and $p_i$'s are all odd, $N=2$.

Question: Is there a general formula for computing $N$ as a function of $p_1,\ldots,p_n$ and $n$?

By the way, though this question is from topology, I feel it more like a combinatoric question and thus have tagged it accordingly.

Answer 1

Answer: When all $p_i$'s are odd, $N=\dfrac{3-(-1)^n}{2}$; when at least two $p_i$'s are even, $N$ equals to the number of even $p_i$'s.

This generalises the condition from Wikipedia about when a Pretzel link is a knot.