(1+2i)x+1 sequence

[Marc LeBrun]

 >=Kohmoto
 >  I considered about an analog of 3x+1 sequence in Gaussian Integer.

Interesting idea.

 > [Definition of 3x+1 sequence]
 > a(n)=(3*a(n-1)+1)/2^k , where 2^k is the highest power of two dividing 
3*a(n-1)+1.

I believe the traditional "Collatz" definition is simpler: "x --> x/2 if x 
even, 3x+1 if x odd".  These are essentially similar, but  I'd recommend 
the simpler form to encourage easier communication.

Then, generalizing with respect to some divisor (2, (1+i), etc), "even" 
just becomes "evenly divisible" while "odd" becomes "not evenly divisible".

 > [A translation to Gaussian integer]
 >
 >  (2+i)x+1 sequence :
 >   S_1  1, 1+2i, 1, 1+2i, ....
 >   S_2  3, 2+5i, 3, 2+5i, ....
 >   S_3  7,  4+11i, 6+7i, 3+10i, ....
 >
 >   (1+2i)x+1 sequence :
 >   T_1  1, 1, 1, 1, ....
 >   T_2  3, 2+3i, 2+5i,  1+8i, 6+i, ....
 >   T_3  5, 3+10i, 1, 1, ....
 >
 > I  am not sure if they are correct, because the factorization is 
difficult without a computer.

This illustrates why simpler definitions might be better.  To simply divide 
x by (1+i) we need only to multiply it by (1-i)/2.  We can then do this 
until x becomes "not evenly divisible".  This is easy to test, since if 
x=(a+ib) then x/(1+i) is (a+b)/2+i(a-b)/2 so we can quickly tell by looking 
at a and b, we don't need to perform difficult factorizations (although a 
computer is still handy).

We thus just need to specify when x/(1+i) doesn't divide out "evenly".  Now 
obviously when a and b have opposite parity the ratio gives half-integers, 
so we at least know that all such numbers are "odd" (although there may be 
others, see next comment).

 > Numbers are calculated in the first quadrant of Z[i] plane.

And this is true just when a>=b.

However I don't think factorization of Gaussian integers is usually defined 
this way.  Instead the "irreducible" factors are taken to lie in the 
"tilted" quadrant between x=y and x=-y containing the positive 
x-axis.  This produces the simple traditional conjugate factorizations such 
as 2=(1+i)(1-i).

Both definitions might be interesting, but we're proliferating generalizations.

 > Do  S_3 , T_2   become periodic?

I don't know, since it depends on what choices you want make in defining 
these things.  However I will say that my quick computer-aided calculations 
seemed to get quite different values than the ones you sent, no matter 
which variations I chose.  They all looked like they were diverging, but I 
didn't have time to dig deeper.

I'd suggest re-analyzing this starting with the simpler definitions, 
eliminating as many cases as possible a priori, to see if there's a "core" 
that are worth exploring via more extensive calculations.

An interesting diagram might be produced by drawing the vectors connecting 
each Gaussian integer with its successor.