[Coda]

That's not really the right way to go about it. It's good that you're developing those insights, but it doesn't quite generalize -- your conclusions are too specific to be especially useful.

A better strategy is to look at the changes in the sequence.

If your sequence is 1 1 1 1 1 (or any constant number), then there are no n's.

Otherwise, find the differences between the steps. 1 2 3 4 5 -> 1 1 1 1, for example, or 1 3 6 10 15 -> 2 3 4 5. Repeat this process until you get to a constant number. Each time you go through this process, the order of the polynomial increases by one -- that is, 1 2 3 4 5 has an n¹ term, 1 3 6 10 15 has an n² term, and 1 16 19 39 69 91 has an n³ term (if I didn't screw up my math). And it sometimes helps to start from 0 instead of 1.