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.