I'm not sure how I missed this before, but \\(f(\frac{a}{b}) = \frac{b}{c}\\) implies \\(f(f(\frac{a}{a+b})) = \frac{b}{b+c}\\). For example, in the 4th row \\(\frac{1}{4}\\) goes to \\(\frac{4}{3}\\) in one step, and in the next row \\(\frac{1}{5}\\) goes to \\(\frac{4}{7}\\) in 2 steps. Proving this would go a long way towards explaining the doubling behavior.