  In 1999, Neil Calkin and Herbert Wilf wrote "Recounting the rationals" which
gave an explicit bijection between the positive integers and the positive
rationals. We find several different (some new) ways to construct this
enumeration and thus create pointers for generalizing. Next, we use circle
packings to generalize and find two other enumerations. Surprisingly, the three
enumerations are all that are possible by using this technique. The proofs
involve, among other things, "negative" continued fractions, Chebyshev
polynomials, Euler's totient function, and generalizations of Stern's diatomic
sequence. Finally we look at some of the remarkable similarities -- and
differences -- of these sequences.
