Hey mathematicians, would you like to help me find patterns between rolling ball clocks?

I recently wrote software to model a rolling ball clock. You can see what a rolling ball clock is here: http://www.youtube.com/watch?v=-3hNCinIBdk and read a description here: http://en.wikipedia.org/wiki/Rolling_ball_clock

The problem I was solving was to figure out how many days it takes for the balls to return to their original order once you start the clock, starting with a given number of balls. Here’s a list of the results: https://gist.github.com/phaedryx/b6ce1df06714c8e66634

I’m wondering if there is some kind of pattern between them, for example, if I know the results of a 27-ball rolling ball clock does that tell me anything about the 54-ball rolling ball clock? Could I calculate a subset of the clocks and extrapolate the results for the rest of the clocks somehow?

I’ve looked at it a bit, but it’s beyond my ken.