Earlier quoted context omitted.
I think this is getting at a pretty good point. Half of all numbers (all evens) will scale by .5 to the subsequent number. The other half (odds) will effectively scale by approximately 3/2, as you have shown here. Obviously over a large amount of iterations this should cause the calculations to trend lower and lower.
One difficulty with this approach is that it doesn't help disproving the existence of cycles, it only makes it less likely that the sequence diverges to infinity.
Another approach would be using the "inverse" of the formula and then trying to prove that you can build all natural numbers starting from 1.
This approach, like every other approaches, did not yield any significant result.
The funny thing is that it fits right in the P=NP problem: easy to verify, hard to solve :)