[seqfan] corollary to the Collatz conjecture?

[Bob Selcoe]

Hi Seqfans,

I'm wondering if there is a corollary to the Collatz conjecture for k >= 0, that 3x + 3^k eventually terminate with a 4-term loop [3^k => 4*3^k => 2*3^k => 3^k].  

So 3x+1 is simply k=0 with terminal loop [1,4,2,1];  k=1 is 3x+3 with loop [3,12,6,3];  k=2 is 3x+9 with loop [9,36,18,9]; etc. 
 
There do not appear to be any OEIS entries pertaining to k > 0.

 I don't know how to program so I can't systematically test the conjecture, but doing some sequences by hand using this online tool http://www.dcode.fr/collatz-conjecture suggests it may hold.

Does anyone know if this has been tested, or if there are any papers specifically addressing the question?

Cheers,
Bob Selcoe