Let $T$ be a Turing machine (or another type of more suitable machine, I am not very confident with this field) with $n$ states and assume that, when started on a blank tape, $T$ does not halt. Interpret the sequence $(x_k)$ of its states (not the number it writes!) as a "decimal" number in basis $n$ (for simplicity, we could assume $n=10$).
(1) Is there any constraint on $(x_k)$? In particular, can it describe an irrational or even a transcendental number?
(2) Could this approach be useful in proving some results about irrationality / transcendence of numbers? (For instance, proving that a number is not irrational by describing it as $(x_k)$ for some Turing machine)