I have had another look at this and arrived at tthe following:

All even numbers will reduce eventually to an odd number. Hence we don't need to consider even numbers at all.

This leaves us with odd numbers. Odd numbers can be split into 2 types: 4n+1 and 4n-1.

The 4n+1 type has the following sequence

4n+1 --> 12n+4 --> 6n+2 --> 3n+1

This last number in the sequence is smaller than the first number - 4n+1 > 3n+1
So we can ignore 4n+1 numbers in our analysis because it will reduce to either another 4n+1 number (in which case it will reduce further) or it will reduce to a 4n-1 type.

Therefore the only types of odd number we need to consider are 4n-1 types.

eg 3, 7, 11, 15, etc.

I have some further analysis (eg 16m+7 always reduces when m is odd...), but I won't post just yet until I have more.