It looks very much like all sequences starting from an n_odd number contain the sequence for (n-1)_odd in the tail, and n = 3 ends in 4,2,1,4,2,1.

At some point a sequence starting from n_even will either reach an odd number part way through, and hence end in 4,2,1,4,2,1 by the above idea, or it will remain even and hence end in 4,2,1,4,2,1. E.g. n=8. = > 8, 4, 2, 1, 4, 2, 1,...

This looks vulnerable to induction.

It looks very much like all sequences starting from an n_odd number contain the sequence for (n-1)_odd in the tail, and n = 3 ends in 4,2,1,4,2,1.

At some point a sequence starting from n_even will either reach an odd number part way through, and hence end in 4,2,1,4,2,1 by the above idea, or it will remain even and hence end in 4,2,1,4,2,1. E.g. n=8. = > 8, 4, 2, 1, 4, 2, 1,...