OEIS/3x+1 Levels

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Attachment rules

The following table (T4) tells the computation rules for the target position, depending on the modularity condition of the compressed source segment, and listed by increasing column number. We identify and denote these attachment rules by the target column number. We show the first segments (their left side) for k = 0, 1, 2, 3. The formulas are rearrangements of the formulas in table T1.

Rule / column	Source segments	First source segments	Target segments	First target segments	Dir.
5	20*(4*k + 3)	3, 7, 11, 15	30*k + 1	1, 2, 3, 4	<
6	20*(4*k + 1)	1, 5, 9, 13	31*k + 1	1, 4, 7, 10	<
9	21*(4*k + 1)	2, 10, 18, 26	31*k + 1	1, 4, 7, 10	<
10	21*(4*k + 3)	6, 14, 22, 30	32*k + 7	7, 16, 25, 34	>
13	22*(4*k + 3)	12, 28, 44, 60	32*k + 7	7, 16, 25, 34	<
14	22*(4*k + 1)	4, 20, 36, 52	33*k + 7	7, 34, 61, 88	>
17	23*(4*k + 1)	8, 40, 72, 104	33*k + 7	7, 34, 61, 88	<
18	23*(4*k + 3)	24, 56, 88, 120	34*k + 61	61, 142, 223, 304	>
21	24*(4*k + 3)	48, 112, 176, 240	34*k + 61	61, 142, 223, 304	>
22	24*(4*k + 1)	16, 80, 144, 208	35*k + 61	61, 304, 547, 790	>
...	...	...	...	...	...

Attachment process

We want to combine all segments such that they form a single tree with root 4 (or 1 in the compressed case). We define two sets:

• The enrooted set E enumerates all segments for which it is still unknown how they should be attached to other segments. E contains all segments from the segment directory in the beginning.
• The disrooted set D' enumerates all segments which have a known attachment rule, target segment and column where they can be attached. D is empty in the beginning.

We now proceed by describing a process which attaches source segments from some subset of E to D. In each step, we

• consider the source segment which are still in E,
• and which have some property,
• and delete those in E and add them to D.

Each step will reduce the number of segments remaining in E. The process does not care whether the target segment is still in E, or already in D.

E is infinite, so such reductions may not help much at first sight. Our goal is, however, to find subsets in E which can be "summed up", that is we look for infinite chains of segments s_i1, s_i2, s_i3, ... where any s_i can be attached to the previous one. For such chains we could move all segments except the s_i1 from E to D.

Segments lengths

The length of a compressed segment is the number of nodes in its right part. The following lengths occur:

• 1 if the segment is constructed by a single µµ operation only, for left sides i ≡ 0, 2 mod 3 - a short segment. Such segments are targeted by rule 5 only.
• 3 if the segment is constructed by the 3 operation sequences µµ, µµδ, µµσ only, for left sides i ≡ 1 mod 3 - a medium segment. Such segments are targeted by rules 5 and 6 only.
• 5, 7 ... otherwise, also for left sides i ≡ 1 mod 3 - a variable segment. Such segments are targeted by rules >= 5. Rules >= 9 always target such a variable segment.

Medium and variable segments are longer than short ones.

Short segments attach to longer segments

In a first step of the attachment process we show that all short segments can be moved from subset E to D since they can be attached to a longer segment:

• Source segments with rules 6 and higher are attached to target segments 3*m + 1 which are longer by definition.
• Rule 5 attaches segments with LS = 4*k + 3 to target segments k + 1. We distinguish 3 cases:
  • For k = 3*m the target is 3*m + 1 and therefore longer.
  • For k = 3*m + 1 the source has LS = 4*(3*m + 1) + 3 = 12*m + 7, which is not short, so rule 5 never applies to these.
  • For k = 3*m + 2 the target is 3*(m + 1), to which we attach the source segment, and make it the new source 3*n. Then we look for the next target and find that either:
    • a rule >= 6 applies which leads to a longer segment, or
    • rule 5 is applicable for a source 3*n which also leads to a longer segment.

In total, E no longer contains short segments, since they were all disrooted and moved to D.

Medium segments attach to a segment in D or a variable one

We examine left sides i ≡ 1 mod 3, but we exclude segment (the root) for the moment. We are concerned with the cases where rules 5 or 6 are applicable, since otherwise the target is variable. As above, we start with rule 5:

i ≡ 1 mod 3 and i = 4*k + 3 and i > 1 => i ≡ 7 mod 12 = 7, 19, 31, 43 ... 
=> k = 1, 4, 7, 10 ... => targets 2, 5, 8, 11 ... => n ≡ 2 mod 3, 

Therefore these targets are short and already in D.

For rule 6 we have:

i ≡ 1 mod 3 and i = 4*k + 1 and i > 1 => i ≡ 1 mod 12 = 13, 25, 37, 49 ... 
=> k = 3, 6, 9, 12 ... => targets (3*k + 1) = 10, 19, 28, 37 ... => n ≡ 1 mod 9 and n > 1

I rule 5 applies for this target, then it is already in D. If some rule >= 9 applies, the claim of this section is also true. We are left with the cases where rule 6 is applicable again:

i ≡ 1 mod 9 and i = 4*k + 1 and i > 1 => i ≡ 1 mod 36 = 37, 73, 109, 145 ... 
=> k = 9, 18, 27, 36 ... => targets (3*k + 1) = 28, 55, 82, 109 ... => n ≡ 1 mod 27 and n > 1

Degrees of nodes

The degree of a node is the maximum number of possible nestings of the form 6*i - 2 in the number:

• 0 if the number is not of the form 6*i - 2 (white),
• 1 if the number is of the form 6*i - 2 (yellow),
• 2 if the number is of the form 6*(6*i - 2) - 2 (orange),
• 3 if the number is of the form 6*(6*(6*i - 2) - 2) - 2 (light red),
• 4 if the number is of the form 6*(6*(6*(6*i - 2) - 2) - 2) - 2 (dark red).

In the segment directories we use warm colors to indicate the degrees.

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