Is there any hope to use a computer to guess combinatorial formulas for a sequence of integer values, given some initial terms?
An exact solution to this problem is of course impossible (for example is $3,5,7$ the sequence of odd integers or of primes?) but there are ways to get answers when the sequence is not in oeis but its "related" to some sequence in the database.

One way to do so is to use superseeker. Basically you send the sequence that you got to superseeker@oeis.org and then the server tries some algorithms in order to find a relationship between your sequence and the ones in oeis.

Hence a good start is to check what kind of tests/algorithms are used by superseeker (iirc there is a description about that somewhere on the site)