You have a couple options when you only have one stud handy.
The easiest is to use a mount similar to this one that mounts on a single stud vertically.
I used this one for both a 22" in the bedroom and a 40" in the living area.
Another option I have seen is if there are a couple of studs which you can mount a finished piece of lumber like 1" x 6" by whatever length you need and then you mount the TV mount on that. Generally your going into a 1" think piece of stud with a 1/16" or 1/8" piece of luan so you can't use more than 1.25" screws unless your mount is thick or you go through the other side of the wall.
Plenty strong of you get good bites with your install screws. I did put a bungee on it when traveling to keep it from extending and putting enough leverage on the mount to pull the screws loose.