OEIS/3x+1 Levels: Difference between revisions
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==Attachment rules== | ==Attachment rules== | ||
The following table '''(T4)''' tells the computation rules for the target position, depending on the modularity condition of the 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 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''. | ||
<!--Generated with | <!--Generated with | ||
<a href="https://github.com/gfis/fasces/blob/master/oeis/collatz/attachtab.pl" target="_blank">segment.pl</a> | <a href="https://github.com/gfis/fasces/blob/master/oeis/collatz/attachtab.pl" target="_blank">segment.pl</a> | ||
at 2019-08-09 | at 2019-08-09 12:09:03;--> | ||
{| class="wikitable" style="text-align:left" | {| class="wikitable" style="text-align:left" | ||
|- | |- | ||
!Rule /<br>column!!Source<br>segments | !Rule /<br>column!!Source<br>segments !!First source<br>segments!!Target<br>segments!!First target<br>segments!!Dir. | ||
|- | |||
|'''5'''||2<sup>0</sup>(4k + 3) ||3, 7, 11, 15||3<sup>0</sup>k + 1 ||1, 2, 3, 4||< | |||
|- | |- | ||
|''' | |'''6'''||2<sup>0</sup>(4k + 1) ||1, 5, 9, 13||3<sup>1</sup>k + 1||1, 4, 7, 10||< | ||
|- | |- | ||
|''' | |'''9'''||2<sup>1</sup>(4k + 1) ||2, 10, 18, 26||3<sup>1</sup>k + 1||1, 4, 7, 10||< | ||
|- | |- | ||
|''' | |'''10'''||2<sup>1</sup>(4k + 3) ||6, 14, 22, 30||3<sup>2</sup>k + 7||7, 16, 25, 34||'''>''' | ||
|- | |- | ||
|''' | |'''13'''||2<sup>2</sup>(4k + 3) ||12, 28, 44, 60||3<sup>2</sup>k + 7||7, 16, 25, 34||< | ||
|- | |- | ||
|''' | |'''14'''||2<sup>2</sup>(4k + 1) ||4, 20, 36, 52||3<sup>3</sup>k + 7||7, 34, 61, 88||'''>''' | ||
|- | |- | ||
|''' | |'''17'''||2<sup>3</sup>(4k + 1) ||8, 40, 72, 104||3<sup>3</sup>k + 7||7, 34, 61, 88||< | ||
|- | |- | ||
|''' | |'''18'''||2<sup>3</sup>(4k + 3) ||24, 56, 88, 120||3<sup>4</sup>k + 61||61, 142, 223, 304||'''>''' | ||
|- | |- | ||
|''' | |'''21'''||2<sup>4</sup>(4k + 3) ||48, 112, 176, 240||3<sup>4</sup>k + 61||61, 142, 223, 304||'''>''' | ||
|- | |- | ||
|''' | |'''22'''||2<sup>4</sup>(4k + 1) ||16, 80, 144, 208||3<sup>5</sup>k + 61||61, 304, 547, 790||'''>''' | ||
|- | |- | ||
|...||...||...||...||...||... | |||
|- | |- | ||
|} | |} | ||
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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: | 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 '''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. | * 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 steps which attach some subset of E to another subset of D. Each step will reduce the number of segments remaining in E. E in infinite, so such reductions may not help much at first sight. Our goal is, however, to end up with subsets in E which can be "summed up", that is we want to find chains of compressed segments ''c<sub>1</sub> = 4, c<sub>i1</sub>, c<sub>i2</sub>, c<sub>i3</sub>, ... '' where any ''c<sub>i</sub>'' can be attached to the previous one. | We now proceed by describing steps which attach some subset of E to another subset of D. Each step will reduce the number of segments remaining in E. E in infinite, so such reductions may not help much at first sight. Our goal is, however, to end up with subsets in E which can be "summed up", that is we want to find chains of compressed segments ''c<sub>1</sub> = 4, c<sub>i1</sub>, c<sub>i2</sub>, c<sub>i3</sub>, ... '' where any ''c<sub>i</sub>'' can be attached to the previous one. | ||
===Attachment of short segments=== | ===Attachment of short segments=== | ||
A compressed segement is '''short''' if it is built by a ''µµ'' operation only (they have index ''i ≡ 0, 2 mod 3), while all other segments (''i ≡ 1 mod 3) are '''long'''. | A compressed segement is '''short''' if it is built by a ''µµ'' operation only (they have index ''i ≡ 0, 2 mod 3''), while all other segments (''i ≡ 1 mod 3'') are '''long'''. | ||
In a first step we show that all short segments can be moved from subset E to D since they can be attached to a long segment. | In a first step we show that all short segments can be moved from subset E to D since they can be attached to a long segment. | ||
---- | ---- | ||
< previous part: [[OEIS/3x%2B1_Connectivity]] ^ up: [[OEIS/3x%2B1_Problem]] | < previous part: [[OEIS/3x%2B1_Connectivity]] ^ up: [[OEIS/3x%2B1_Problem]] | ||
Revision as of 10:11, 9 August 2019
< previous part: OEIS/3x+1_Connectivity ^ up: OEIS/3x+1_Problem
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.
| Rule / column |
Source segments |
First source segments |
Target segments |
First target segments |
Dir. |
|---|---|---|---|---|---|
| 5 | 20(4k + 3) | 3, 7, 11, 15 | 30k + 1 | 1, 2, 3, 4 | < |
| 6 | 20(4k + 1) | 1, 5, 9, 13 | 31k + 1 | 1, 4, 7, 10 | < |
| 9 | 21(4k + 1) | 2, 10, 18, 26 | 31k + 1 | 1, 4, 7, 10 | < |
| 10 | 21(4k + 3) | 6, 14, 22, 30 | 32k + 7 | 7, 16, 25, 34 | > |
| 13 | 22(4k + 3) | 12, 28, 44, 60 | 32k + 7 | 7, 16, 25, 34 | < |
| 14 | 22(4k + 1) | 4, 20, 36, 52 | 33k + 7 | 7, 34, 61, 88 | > |
| 17 | 23(4k + 1) | 8, 40, 72, 104 | 33k + 7 | 7, 34, 61, 88 | < |
| 18 | 23(4k + 3) | 24, 56, 88, 120 | 34k + 61 | 61, 142, 223, 304 | > |
| 21 | 24(4k + 3) | 48, 112, 176, 240 | 34k + 61 | 61, 142, 223, 304 | > |
| 22 | 24(4k + 1) | 16, 80, 144, 208 | 35k + 61 | 61, 304, 547, 790 | > |
| ... | ... | ... | ... | ... | ... |
Enrooted and disrooted sets
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 steps which attach some subset of E to another subset of D. Each step will reduce the number of segments remaining in E. E in infinite, so such reductions may not help much at first sight. Our goal is, however, to end up with subsets in E which can be "summed up", that is we want to find chains of compressed segments c1 = 4, ci1, ci2, ci3, ... where any ci can be attached to the previous one.
Attachment of short segments
A compressed segement is short if it is built by a µµ operation only (they have index i ≡ 0, 2 mod 3), while all other segments (i ≡ 1 mod 3) are long.
In a first step we show that all short segments can be moved from subset E to D since they can be attached to a long segment.
< previous part: OEIS/3x+1_Connectivity ^ up: OEIS/3x+1_Problem