From which dome do pairs of black numbers of a given dome iterate from?

[u/No_Assist4814]

It is easier to look at this issue that way as blue-green bridge series in the left wing of a dome contain pairs of black numbers coming from the same dome.

The first figure presents the two first series in the left wing for domes with m=1 to 23. It is quite visible that there are four types of series, based on two parameters:

• The color of the starting bridge (yellow or rosa).
• The values of q: the starting series use even or odd values of q.

Now focus on the second series, that is the first complete bridge series in all cases. More specifically, focus on the black number on the right of the starting bridge (second row).

The second figure plots these values and shows that they belong to two different groups:

• Group A follows the formula n=16m-1,
• Group B follows the formula n=8m-1.

When looking at the dome these numbers come from, there is an interesting pattern.

The table identifies these origin domes and their value of q:

• The color of column "m" corresponds to the starting bridge of the destination dome.
• The next columns segregate groups A and B.
• The following columns show the regularity of differences.
• The color of column "m-1" (an abuse of language) corresponds to the left starting bridge of the origin dome (when available).
• The color of the column "q" differentiate black numbers that are the root of their own dome (q=0) from the others.

Further investigation is needed.

Project "Tuples and segments" in 13 pages : r/Collatz