Some sequences have conjectured formulas, for example the following which I encountered recently, [A227404][1].

$a_n$ is the total number of inversions, among all permutations on $[n]$ that is a single cycle in the cycle decomposition.
It is conjectured that $a_n = n! (3n-1)/12$, and I find it rather amazing that there is no proof of this.


  [1]: https://oeis.org/A227404