I regret my question came across as vague. Thank you very much for this reply, I am very glad to have your answer to study. When asking the same question, I was advised on Reddit that "Every infinite periodic continued fraction gives rise to a quadratic equation, one of whose roots is the value of the fraction" and that this positive root might "be expressed using a recurrence, in exactly the same way that you can get continued fraction convergents using a recurrence." I was aware of how CFs can be truncated and their convergents given an equation, but as you noted, I wanted to study the infinite values, before I consider such truncations. Your answer has made a lot of sense to me (phew!) and I'm sure it will continue to do so more as I study it. Thank you again.