For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the same mark, which suggests a connection between a priori independent mathematical areas. The most famous example like that is perhaps the Catalan numbers sequence: A000108.
Question: What are the examples of pair of integer sequences coinciding on all the known terms, but for which the coincidence for all the terms is unknown?
Cheating is not allowed. By cheating I mean artificial examples like:
$u_n = v_n =n$ for $n \neq 10$, and if RH is true then $u_{10} = v_{10} = 10$, else $u_{10}+1 = v_{10} = 1$.
The existence of an OEIS entry could act as safety.
EDIT: I would like to point out that all the answers below are about pair of integer sequences which were already conjectured to be the same, and of course they are on-topic (and some of them are very nice). Note that such examples can be found by searching something like "conjectured to be identical" on OEIS, as I did for some of my own examples below…
Now, a more surprising kind of answer would be a (non-cheating) pair of integer sequences which are the same on the known entries, but for which there is no evidence a priori that they are the same for all the entries or that they are related (i.e. the precise meaning of a coincidence). Such examples, also on-topic, could reveal some unexpected connections in mathematics, but could be harder to find…
Best Answer
A historical example, in the sense that the conjectural equality has been refuted: A180632 (Minimum length of a string of letters that contains every permutation of $n$ letters as sub-strings) was conjectured equal to A007489 ($\sum_{k=1}^n k!$).
The exact value of A180632 at $n=6$ is still unknown, but it must be less than the conjectured value of $1!+2!+\cdots+6!=873$, because the following string of length 872 contains every permutation of 123456 as a substring: