Probably. But I find it difficult to will myself to go to sleep, so I can't use a timer; I guess I could try using something on my phone

This is important. I've found that 22.5 x 2^k min for any reasonable k has me waking up feeling refreshed.

Wait, what? These numbers seem off..

k = 1 => 45 minutes

k = 2 => 1.5 hours

k = 3 => 3 hours

k = 4 => 6 hours

k = 5 => 12 hours

That's exactly it (except that k=5 is not reasonable). The REM periods get longer and longer as you sleep; that is, the "cycles" do not have the same period.

Edit: also, k=0 is valid, and what most people have been advocating here (as 20 minutes).