Hey pitfer.

There are ways to derive sequences (like recurrence relations) but one of the down-falls to this approach is that you can fit an infinite number of sequences to a finite list just like you can pick an infinite number of polynomials have the same roots (for the polynomial case, you just change the multiplicity of the roots).

For this reason even if you did get a sequence, it would be only one of many.

Although I am not an expert in the subject, I do know that there are techniques and algorithms to find sequence definitions but as I said above, I don't know if they will really help you all that much due to the nature that is sequences where you only have a small subset of the entire sequence.