Behavior of the rational transformation a/b → (b−a)/a → b/(b−a) → a/b

Let a/b be any rational number where b > a and |b−a| > 1. I found that applying the transformation

a/b → (b−a)/a → b/(b−a) → a/b

returns to the original value after exactly 3 steps. This lets any such a/b be written as an infinite nested fraction (a form of continued fraction):

$\frac{a}{b} = \cfrac{1}{1 - \cfrac{1}{1 - \cfrac{1}{1 - \frac{a}{b}}}}$

I verified this for 7/10 and 3/10 numerically.

However, I found that not all rational numbers follow this particular transformation. Specifically, rational numbers of the form (2a+1)/a² behave differently.

• For a = 6: 13/36 does not return to itself after 3 steps.
• For a = 5: 11/25 behaves differently from the a=6 case.
• For a = 8: this case also does not follow the pattern.

When I traced the full chain starting from 13/36, I got:

13/36, 27/13, 10/23, 17/10, 7/13, 10/3, 7/10, 3/7, 10/3, 7/10, ...

This eventually enters a repeating loop: 7/10 → 3/7 → 10/3 → 7/10 → ...

I noticed that for the first 7 terms of this chain, the value alternates strictly between less than 1 and greater than 1: 13/36 (<1), 27/13 (>1), 10/23 (<1), 17/10 (>1), 7/13 (<1), 10/3 (>1), 7/10 (<1)

This alternation breaks exactly at the point where the chain enters the repeating 3-cycle (7/10 <1, then 3/7 <1 — two "less than 1" terms in a row).

I also tested rational numbers of the form (3a+1)/a³. For a = 6, I did not get a similarly clean result.

My questions:

1. Why do rational numbers of the form (2a+1)/a² fail to return to themselves under this transformation, while other rationals like 7/10 do?
2. Is there a known characterization of which rationals a/b return to themselves in exactly 3 steps under this transformation, versus which are eventually periodic (feeding into a different cycle)?
3. Is there a reason the alternation (above/below 1) breaks exactly at the point of entering the repeating cycle?