My "regular" breathing pattern (i.e. the one I use to run at medium-hard paces for long distances) is:
nose-in, mouth out: in, out-out, in-in, ooout.
1/4, 1/8, 1/8, 1/8, 1/8, 1/2.
I attenuate the BPM to my pace: on high intensity runs, I usually can get away with raising the BPM, but the last 1/2 is very hard and I sometimes break it into 1/4 out, 1/4 in (more oxygen). For sprinting, any pattern goes away, and it's 1/4 in, 1/4 out. Still nose to mouth, but I think the optimal solution here depends a lot on the shapes of your nose/mouth. :)
And most people think running is just putting one foot in front of the other! :)
To some extent it is of course, but these kinds of cadences are necessary to run effectively, which I guess is different from just running.