i dont think a pattern can be established with only two symbols by definition of a pattern.
For example, given two points on a missile trajectory can you predict the third point. You'd have to assume a line but the missile would most likely be on an arc so given a third point gives you enough info to predict the fourth and beyond with ever greater precision.
Now in statistics, they often look for correlations between two variables and that is a kind of pattern.
Well, you can extend the applicability of a pattern to as many terms as you like, so ... there is no such thing as a finite pattern in the sense it is commonly construed.
On the other hand, regarding periodicity, there has to be some underlying frequency in tabulation as determining the nature of the series in question. Otherwise, there would be no pattern.
If there is something known and recognized as a pattern, it is only so because it is periodical. Even in chaos theory, in deriving a new Xp from a given Xp-1, we follow certain rules as overlapping our initial states.
On another tangent, a truly non-periodical pattern, it would appear, seems demanding of an infinite regress wherein irony is delivered to the persistent. Something like that cannot even conceptually exist in understanding, at least not to my current awareness nor concern.