another pattern for 3x+1 in base 4

[Gottfried Helms]

Writing numbers in base 4. The (to xxx) parentheses could even better
be written in base 64... I've shown this already, but not in this
concise scheme.


Numbers, that fall down to 1 in a certain number of steps.

Where one step is:  

    compute from odd n0 the next n1
      n1 = (n0 *3+1)/2^k
    with k the highest possible integer which leaves the whole term integer.


The dots in the following numbers are inserted only for better readability.

in 1 step to 1 --------------------------------------------

  (to 1)            1,   1.1,  1.11,   1.111,   1.1111,...
                               (=1,5,21,85,341,... <base 10>) 


in 2 steps to 1 -------------------------------------------

[ (to 1)            1,         1.1,         1.11,        1.111,...]
  (to 1.111)        1.301,     1.301.1,     1.301.11,     ...
  (to 1.111111)     1.301.301, 1.301.301.1, 1.301.301.11, ...
  ...
  (to 1.1)          3,         3.1,         3.11,        3.111,...
  (to 1.1111)       3.203,     3.203.1,     3.203.11,     ...
  (to 1.1111111)    3.203.203, 3.203.203.1, 3.203.203.11, ...
  ...
  (to 1.11)         -none-
  (to 1.11111)      -none-
  ...

in 3 steps to 1 -----------------------------------------

 (via 1.1) -------------

  (to 3)            -none-
  (to 3.111)        -none-
  ...
  (to 3.1)          101,     101.1,     101.11,     101.111,...
  (to 3.1111)       101.301, 101.301.1, 101.301.11, ...
  ...
  (to 3.11)         203,     203.1,     203.11,     ...
  (to 3.11111)      203.203, 203.203.1, 203.203.11, ...
  ...
  


 (via 1.1111) -----------

  (to 3.203)        2113,     2113.1,      2113.11,...
  (to 3.203.111)    2113.203, 2113.203.1,  2113.203.11,...
  ...
  (to 3.203.1)      -none-
  ...
  (to 3.203.11)     1023.301,     1023.301.1,     1023.301.11,...
  (to 3.203.11111)  1023.301.301, 1023.301.301.1, 1023.301.301.11,...

...

 (list all remaining of step 2 here analogously)

 --------------------------------------------------------

These lists are all connected via the 3x+1-operation. Unfortunately ;-)
it is still not obvious, that or how they are exhaustive for all odd natural
numbers (but what did you expect?)

Any good way to put these lists in a sequence (besides 1,2,3,4,5...)?

Gottfried Helms