I wonder what are the examples of integer sequences, where only few elements are known and the researchers are still actively looking for the new terms. I think this discussion might be a good reference for those who would like to contribute to mathematical community by doing large scale calculations, extending the numerical data and providing computational evidence towards known conjectures.
The examples should include only those sequences where it is potentially feasible to compute one of the unknown terms if one has an optimized algorithm and enough computing power. In other words, the problem should be more computational rather than theoretical.
Perhaps, the most famous example are Mersenne primes (A001348), where only the first 45 consecutive terms are known (even though 49 Mersenne primes are known):
$3, 7, 31, 127, 2047, 8191, 131071, 524287, 8388607, 536870911, 2147483647, \ldots$
The members of GIMPS (Great Internet Mersenne Prime Search) are actively searching for the new terms, and according to the website are currently trying to prove that $M_{37156667}$ is the 46th Mersenne prime.
Another interesting example is the number of arrangements of $n$ circes in the affine plane (A250001), where only the first 5 terms are known:
$1, 1, 3, 14, 173.$
All of these elements got computed by Jon Wild, and on the OEIS webpage for this sequence he mentioned that the 5th element (the sequence starts with the 0th term) is probably equal to 16942.
I'd be interested to see other curious examples.
Best Answer
I think the question is too broad. There are a variety of such examples, and it seems that many combinatorial items and computational complexity items would fit the bill. The OEIS has a "most wanted" list, and so do I, but it is not clear which of these are a best fit for your question.
I start with http://oeis.org/A051236 which relates to the determinant spectrum problem mentioned by Will Orrick in https://mathoverflow.net/a/21510. Even conjectured values are known only for the first twenty or so terms. There are many related sequences, including http://oeis.org/A003432, the maximal determinant of an order n real 0-1 matrix. These are computationally challenging because we only know what such a matrix looks like for some n. We do not understand the determinant function as a combinatiorial entity enough to predict its range on certain sets.
Ramsey numbers provide several easily estimated and computationally hard sequences. I'll let an expert expound on these.
http://oeis.org/A005245 is another favorite of mine relating to integer complexity. Harry Altman has studied this, and posted on MathOverflow about it. Many terms can be computed now, but since for each n it involves a minimum over exponentially many configurations, one has to be clever in how to compute it. It is less challenging than computing Ramsey numbers, but you can't analyze it and compute bits of it like you can a Fibonacci sequence.
There is Dedekind's problem, http://oeis.org/A000372, or for universal algebraists, the size of the free distributive lattice generated by n elements. In fact, free spectra and enumeration of structures satisfying certain not necessarily equationally expressed conditions are subjects of ongoing research. Union closed families fall under http://oeis.org/A102897, and are another personal favorite.
Under suitable encoding/listings, various sequences of 0's and 1's are of interest. These are languages of varying computational complexity, usually corresponding to whether the nth structure of a list has property P or not, for example if a given graph has a Hamiltonian cycle.
I think all of these examples and many more fit your criteria. You might consider revising your question to give more examples and nonexamples from the OEIS corpus.
Gerhard "Or Maybe Check Their Website" Paseman, 2017.03.02.