Good Subwoofer recommendations

I'm looking for a good Sub for my existing HT system.

There are so many types like Sealed, Ported, Cylinder, Bass-reflex, etc. & it's a bit confusing to determine which one will be better. Also, does type of speakers or Amps I have can impact the decision?

My budget is around $1000 & I prefer getting a used or refurbished rather than new.

Current setup:
LRC Speakers: Tyler Acoustics Linbrook System II
Surround Speakers: JBL S312
Amplifier: Parasound Halo A51
BD Player (cum PreAmp): Oppo BDP-105
Room: 16x18+ with ~40% open walls, so sound can bleed

Thank you in advance!
There is a Bag End Infrasub 18 for $895. That sub was a review winner many times for HT. Probably can't go wrong with that. With 40% open walls you will need the size as well.
I think SVC makes great lower cost subs. A critical question, though, is if you are just going to be doing movies or will this be your TT/CD/SACD system as well? If you want it to do everything, I strongly recommend a sealed set of subs. Ported subs are just too slow, unless you want to do some serious modification. In my HT room I'm using a set of sealed subs and have found it a fair trade off to the one ported sub that is found in so many rooms. I do have a 15" sub in the garage/gym that is ported, but even there I had to modify the port and use a lot of Marigo speaker dots to get it to be listenable. I have to say, though, exercising to a 15" sub is motivational!
Rythmik has their E15HP in matte black on sale right now for $1024 plus shipping. 15" Servo-Feedback woofer, 600w amp. Best deal in subwoofers.
Check out SVS. They are a great company. 45 day return policy with free shipping. I had the SB-1000 and it was outstanding.
Rhythmik and REL make very good subs, as do SVS. However, my favorite subs are the Velodyne DD and Optimum series - these are *fantastic* subs...

There is a Vandersteen V2w here for $950 that would be my choice. I have the audio-only version (2Wq) which has been a terrific performer--fast, tuneful and integrates well with the speakers.

PSA xs15se 15 inch sealed down fire, excellent for ht and music. I think they are about 750 brand new and delivered.
I had Tyler custom-fabricate a really nice subwoofer for me two years ago. Fits the bill perfectly, and the finish matches my Woodmeres.
Is sub primarily going to be for music or HT? if for music I go with REL and not look back. REL is tight, fast and musical. good with HT but not great, unless you go with their higher offerings. I just ordered two REL t9's which houses two 10inch drivers and should fit the bill for my room and should do pretty well with HT.
Thanks for all your suggestions. I ended up buying my friend's 10 years old SVS 20-39PC Plus.

It's not as high-end & versatile as some of the models suggested here but it does the job fairly well at a fraction of the cost ($250). It's 40" tall & has custom-built 12-inch woofer with a 525 watt power amplifier.

Except of it's physical appearance, during playback you'll hardly notice it is even there while still providing fuller & richer sound.

Thank you once again!