Providing a sequence of numbers, with little to no context, and asking for either missing values in the sequence or the rule by which the sequence is generated, is typically a bad puzzle. The reasons are:
- Such a sequence is guaranteed to be non-unique. For any finite sequence of n real numbers, it is possible to create infinitely many polynomials that generate that sequence, in addition to any number of other possible generating functions and/or algorithms.
- If the generating method is sufficiently general and/or well-known, it is likely that the sequence is already listed in a database such as the OEIS.
- If the generating method is sufficiently specific and/or obscure, then with no context there is no way of knowing what kind of process the asker had in mind.
- All of the above, combined, mean that any such question reduces to a guessing game where the asker continually rejects apparently valid but not intended solutions, which is a sign of a poor puzzle.
What's the next number? 1, 2, 4, ???
8 is an obvious answer (powers of 2), but a little searching will show that there are valid arguments for 10 ("left factorials"), or 16 (raise 2 to the previous number in the sequence), or 7 (number of pieces you get when you cut a circle with n lines, or numbers with an odd number of 1s in their binary expression), and if you want to be a dick then you can construct arbitrarily many other solutions.
This is on a par with asking:
What's the missing word? CAT, RAT, DOG, ???
and then rejecting a host of attempted solutions (ANT, MOUSE, OWL, AMOEBA) because they aren't the "right" solution you were thinking of.
Some things that you can do to help improve the question include:
- Providing sufficient numbers in the sequence to reduce ambiguity. While you can't completely remove all alternative ways to generate the sequence, you can make it so that the intended method is immensely simpler than any other.
- Giving additional contextual information to lead towards the intended solution. If the solution relates to the way the numbers are spelled, then at least put in a cryptic "I won't spell it out for you, but ..." or something. If it's about Pascal's triangle let people know "You might have to look at some combinations of things".
- Checking the OEIS to make sure the sequence doesn't already appear there in some form. If you're feeling particularly thorough, use their SuperSeeker email address to check whether your sequence is related to one they have (e.g. your sequence might be the prime numbers squared minus the next prime number, and if that weren't a sequence in their database already - which it is - it would still give you the breakdown of where it comes from).