Are there any techniques for doing this, or is it mainly a intuition and check process?
We can't say like that because some follow a series and some follow logic.I'll give examples of both.Series'll follow a specified formula.
Eg. of logic series:1,2,2,4,3,8,4,16,5,6etc. In this odd positioned nos are 1,2,3etc and even positioned nos are 2,4,8,16 etc.
Sequences of the form f(n) where f is a polynomial function can be detected and analyzed by making "tables of differences" and looking for a row of zeroes in the table. The polynomial can be reconstructed from such a table. Systematic methods for reconstructing sequences based on mathematical formulae are treated in the mathematics known as "The Calculus Of Finite Differences".
As omkar13 indicates, there are sequences not based on simple mathematical expressions. For example you could number the typographical symbols used in a book and make a sequence corresponding to a short story, say "The Library Of Babel" by Jorges Luis Borges. Perhaps there is a mathematical formulae that would produce that sequence of symbols. I don't know. (Just because the author didn't use such a formula doesn't mean there isn't one.)
I understand that nearly all possible series won't have a formula. We've just wrapped up integration and started on these and part of the homework was doing this.
It's a lot of fun, actually, but sonetimes it would take me too long.
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