I've been reading Wikipedia's article on continued fractions.
A few examples are given for the continued-fraction representation of irrational numbers:
The pattern repeats indefinitely with a period of $6$.
The pattern repeats indefinitely.
The pattern repeats indefinitely with a period of $3$, with $2$ added to the 3rd term in each cycle.
The terms in this representation are apparently random.
The first two are algebraic irrational, and the last two are transcendental.
We differentiate between these two types of irrational numbers by definition:
A number is algebraic if and only if it is a root of some non-zero rational-coefficient polynomial.
This gives me the impression that transcendental numbers are "more irrational" than algebraic irrational numbers, so to speak (please excuse me for the non-mathematical notation here).
The continued-fraction examples above give me an additional impression that some transcendental numbers are yet even "more irrational" than others (i.e., $\pi$ is "more irrational" than $e$ and $\phi$).
Has any work been made towards differentiating between those transcendental numbers which can be represented with a "patterned" continued-fraction, and those which cannot?
Moreover, as with algebraic numbers which consist only a "small" countable part of all real numbers (where transcendental numbers consist of the "much larger" uncountable part), is it possible that the "patterned" transcendental numbers are countable and the "random" transcendental numbers are uncountable?
This question can be reduced to a distinction between "patterned" sequences of natural numbers and "random" sequences of natural numbers. I think that the former set is countable and the latter set is uncountable, but I am not sure how it can be proved (mostly because I am not sure how to mathematically differentiate between "patterned" and "random").