I'm not sure how to talk about what I want to talk about, so I'll give some examples.
The number $\pi$ is irrational and has no repeating pattern, but is computable by an easy rule; divide the circumference of a circle by its diameter.
Now consider the number $\sum_{i=1}^{\infty}10^{-(i!)}.$ This has a pattern, and by definition generated by a defined rule. But the number is still irrational.
My question is, is there a mathematical concept similar to, but more general than, rationality that differentiates between these different types of numbers?