I am glad it was short for you, but it varies in general. For some people it doesn't take long. For others, it takes longer. For some, reconnection occurs so it is important to check again 1 year after the vasectomy.
15-20 ejaculations typically clear any remaining sperm, but again none of this is universal and it needs to be tested.
People should be paying you to get a vasectomy. When I got mine (8 years ago now?) it was $800. It is the single most effective thing one can do for the environment, and is a real vote against these tyrannous governments oppressing people (women specifically in the case of banning abortion).
For women, sterilization is involved and dangerous. For men, it is a 5 minute procedure. Do it!
Start taking the steps!
Don't forget that after you get it, you need to wait around 6 months before you're sterile. Then, you need to get your sperm count checked. It is important, the vasectomy is not immediate sterilization.
Erroring on the side of avoiding a pregnancy is a win in my book ๐