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:

https://www.jasondavies.com/collatz-graph/Collatz graph

one immediately see that the graph is symmetric around the central node, each branch having a counterpart opposite from the central node. enter image description here 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.


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    $\begingroup$ This always happens, since given any even $3k+1$, there are two numbers that map to that even, and their sequences behave identically. $\endgroup$ – Don Thousand Dec 13 '19 at 11:39
  • $\begingroup$ @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). $\endgroup$ – pisoir Dec 13 '19 at 12:49
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    $\begingroup$ 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:) $\endgroup$ – pisoir Dec 13 '19 at 12:57
  • $\begingroup$ any number divisible by 3 won't branch further. $\endgroup$ – user645636 Dec 13 '19 at 13:21
  • $\begingroup$ mathoverflow.net/questions/288807/… $\endgroup$ – Collag3n Dec 13 '19 at 17:51

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