"We’ve chosen a rule that some sequences of three numbers obey — and some do not. Your job is to guess what the rule is."

My mistake was to assume that choosing the first number uniquely defines the next ones in the sequence. Since, you know, like all the sequence puzzles I've seen before worked like that, and I didn't read it rigorously enough. Oh, by the way, the doubling thing is wrong if you use negative numbers (wrong as in it gives false positives, instead of just false negatives). But the problem definition doesn't even tell what set of numbers we're operating on.

Finding the rule the sequences obey is impossible since it could be that all cases follow a simple rule except for one triplet which you're unlikely to find. It's trivially easy to fool the user into finding a wrong rule.