# Is the inverse Collatz graph symmetric?

Maybe a stupid question, but I couldn't find a direct answer. When looking at this beautiful visualization of the graph of inverse Collatz sequence:

one immediately see that the graph is symmetric around the central node, each branch having a counterpart opposite from the central node. So I am wondering, is this just an illusion and it eventually breaks down? From what I know about the Collatz sequence I was definitely expecting some more random branching.

Thanks.

• This always happens, since given any even $3k+1$, there are two numbers that map to that even, and their sequences behave identically. – Don Thousand Dec 13 '19 at 11:39
• @DonThousand Ok. I think I mostly understand, but it still surprises me little bit that even the length of "even" sequences are "symmetric" - 853, 1706, 3412 vs 5461,10922, 21844. (both length 3 before it branches again). – pisoir Dec 13 '19 at 12:49
• Ok. It actually does indeed seem to be only an illusion. The branch 11,22 branches after 22 into 7 and 44, while the opposite symmetric branch 75, 150, 300, 600, 1200 does not branch. Which, nevertheless, seems to be the only exception in that graph (as far as I noticed). Very interesting still:) – pisoir Dec 13 '19 at 12:57
• any number divisible by 3 won't branch further. – user645636 Dec 13 '19 at 13:21
• mathoverflow.net/questions/288807/… – Collag3n Dec 13 '19 at 17:51