Collatz vs. rotate

[Marc LeBrun]

Like a moth to a candle, David Wilson's note about Collatz being 
2-automatic reminded me of something... I've wondered if anyone else has 
ever explored anything along these lines?

Obviously since 3n+1 is even when n is odd, we can just as well fold in the 
inevitable halving step, getting f(n) defined by
   2k   --> k
   2k+1 --> 3k+2

The nearly trivial observation is just that f(n) is very similar to the 
"rotate right" function r(n) = A038572:

f(n):  0 2 1 5 2 8 3 11 4 14 5 17 6 20 7 23 8 26 9 29 10 32 11 35 12 38 13 
41 14 44 15 47 16...
r(n):  0 1 1 3 2 6 3  7 4 12 5 13 6 14 7 15 8 24 9 25 10 26 11 27 12 28 13 
29 14 30 15 31 16...
diff:  0 1 0 2 0 2 0  4 0  2 0  4 0  6 0  8 0  2 0  4  0  6  0  8  0 10  0 
12  0 14  0 16  0...

These even positive differences are just twice the sequence

        1, 1 2, 1 2 3 4, 1 2 3 4 5 6 7 8, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 
15 16...

Anyway, while f(n) apparently iterates chaotically to 1, "destroying" all 
the information originally in n, in contrast r(n) preserves the 1 bits, and 
so fixes onto the points 2^k - 1.

This makes it smell to me like information-theoretic approaches to Collatz, 
such as this k-automatic stuff, might indeed be very fruitful.

(Or at least there's some simple sequences I owe the EIS... when I'm more 
awake!<;-)