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.

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.

The kissing numbers give another interesting example (A257479). Only the first 4 consecutive terms in this sequence are known:

$2, 6, 12, 24.$

Note that the 8th and the 24th elements in this sequence are known, too. They are 240 and 196560, respectively, as was proved by Odlyzko and Sloane in 1979.

I'd be interested to see other curious examples.