Actually some time ago I used to always say "17" when somebody asked me how to continue a a given series.

So I thought about a more strict formulation of the question. I came up with something along these lines:

Given a series @a = $a[0] ... $a[$n-1], find all pure functions f($n, @a) that only uses the operators +, -, *, /, % and **, the arguments and number literals (and parenthesis for grouping), and where the following identity holds: For all $k <= $n: f($k, @a[0..$k-1]) == $a[$k].

Of all these functions, use the simplest, ie the one that uses the smallest number of operators. Then calculate the next value with that function.