And, of course, mathematical psuedo-sophisticates make appeals to Kolmogorov complexity which is unhelpfully uncomputable.
What I would like instead are a collection of mechanisms for proving such questions meaningless--- that is, tool for constructing an arbitrary fit for an integer sequence. Here's a few to kick things off:
1. Fit the K integers as successive values x=1, x=2, x=3, ..., x=k of a polynomial of order N. There are N+1 coefficients, so when N+1 >= K we should be able to find an exact fit (except maybe in some degenerate cases not immediately obvious to me) and for N+1 > K we can find an arbitrary number of polynomials.
2. Take N=ceil(log2(max(Ai))). Then we can view the sequence as the operation of N separate N-bit binary functions. Since there are 2^N such binary functions and only K examples, the problem is not particularly constrained, even if we restrict ourselves to "simple" N-to-1-bit functions.
3. Define A_K as a (K-1)-ary function of its previous arguments. Backfill A_0, A_-1, A_-2 to make the sequence look non-arbitrary.
4. Take differences between terms enough times, until the resulting sequence is short enough to be matched to something trivial.
5. Split the sequence into two or more unrelated sequences that have been interleaved (which can be done in a variety of ways.)
6. Construct a rational number such that A1 A2 A3 ... AK (suitably zero padded) are the repeating digits (or initial digits, or a combination) of its decimal expansion.
7. Construct a number such that A1, A2, A3, ... AK are the first terms in its continued fraction representation.
Any other suggestions?