After checking OEIS, the first thing to do is usually to try polynomial extrapolation by the method of differences:
- Calculate the differences between successive numbers
- See if all the results are the same.
- If they are, you have successfully fitted a polynomial to the sequence.
- If they are not, start again, using the differences as the number sequence.
The earlier you hit identical numbers, the more credible your polynomial fit is. If you have, say, three identical digits, you can be almost sure you got the intended solution. On the other hand, if you continue until you only have one number left, you have fitted an Nth order polynomial to N+1 data points, which is always possible, and the extrapolated value will have nothing to do with the intended solution.
If you find a credible match, just add one more of those identical numbers on the right, and work backwards to get the next number in the series.
Now then, if both OEIS and the polynomial fit fail to give an answer, then the situation is called a number sequence puzzle. Because many simple good ideas have already been had, check with the usual suspects first:
- self descriptive sequences
- getting the next number by manipulating the digits of the previous one in some way
- and so on, there are many good ones to be found in the number-sequence tag.
If you still got nothing, then you either have an interesting, original puzzle, or a useless, unsolvable one. You'll need to figure out which one, in order to decide how many hours you are going to spend. There sure have been many gems on this site, but many of the other kind too.