I figured out what to do when k is even. I knew it involved doubling the period or something similar, but it wasn't working out. Sorry for the confusion.
Anyhoo, ionce you get to and k is even, you need to find x, y with , then x, y are your minimal solutions.
The term shouldn't be suprising. If you were at , , and you wanted to find more terms, you'd see . Muck about with a few examples and you'll see why this is so.