With the exception of the second term, it is not hard to prove that the sequence increases monotonically. Empirically, the growth is super-exponential.
For even k, there are no such k-cycles in S_k whose commutator with the standard k-cycle is a k-cycle.
Table of n, a(n) for n=1..6.
Sequence in context: A239651 A014484 A110040 * A181042 A263370 A129866
Adjacent sequences: A298071 A298072 A298073 * A298075 A298076 A298077
Tarik Aougab, Jan 11 2018