Rich Schroeppel asked me:

Neil, have you looked at "Farey-depth"? I.e., 1. the number of mediant steps required to reach a particular rational? Or 2. the number of steps needed to reach the earliest and latest k/N (as a function of N = 1, 3, 5, 7, ...).

This was in response to my comment that

I added the triangles of numerators and denominators of these fractions to the OEIS: see http://oeis.org/A177405 and A177407.

I haven't looked at "Farey-depth". Maybe someone on this list could work out the initial terms of the two sequences in 2.? Neil