Collatz Streetmap: Difference between revisions

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===Introduction===
When all Collatz sequences are read backwards, they form a graph starting with 1, 2 ..., hopefully without cycles (except for 1,2,4,1,2,4 ...). At each node n in the graph, the path starting at the root (4) and  with the last node n can in principle be continued to 2 new nodes by a
* "m"-step: n * 2 (which is always possible), or a
* "d"-step: (n - 1) / 3 (which is possible only if n - 1 mod 3 = 0).
When n mod 3 = 0, the path will continue with m-steps only, since the duplication maintains the divisibility by 3.
===Literature===
* Jeffry C. Lagarias, Ed.: ''The Ultimate Challenge: The 3x+1 Problem'', Amer. Math. Soc., 2010, ISBN 978-8218-4940-8. [http://www.ams.org/bookpages/mbk-78 MBK78]
* [http://oeis.org/A070165/a070165.txt  File of first 10K Collatz sequences], ascending start values, with lengths
* [http://oeis.org/A070165/a070165.txt  File of first 10K Collatz sequences], ascending start values, with lengths
* [http://go.helms-net.de/math/collatz/aboutloop/collatzgraphs.htm The Collatz-Problem]. A view into some 3x+1-trees and a new fractal graphic representation. Gottfried Helms, Univ. Kassel
* [http://go.helms-net.de/math/collatz/aboutloop/collatzgraphs.htm The Collatz-Problem]. A view into some 3x+1-trees and a new fractal graphic representation. Gottfried Helms, Univ. Kassel
* [https://de.wikibooks.org/wiki/Collatzfolgen_und_Schachbrett Collatzfolgen und Schachbrett] auf Wikibooks
* [https://de.wikibooks.org/wiki/Collatzfolgen_und_Schachbrett Collatzfolgen und Schachbrett], on Wikibooks
===Patterns in similiar sequences===
===Motivation: Patterns in sequences with same length===
* 142, 143 with same length (from [https://oeis.org/A070165 A070165]):
When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, and with a characteristical neighbourhood of every other value, for example (from [https://oeis.org/A070165 OEIS A070165]):
  142/104: [142 m  71 d 214 m 107 d 322 m 161 d 484 m  242 m 121 d | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
  142/104: [142 m  71 d 214 m 107 d 322 m 161 d 484 m  242 m 121 d | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
             +1  *6+4    +1  *6+4    +1  *6+4    +1  *6+4  *6+2      =    =  ...
             +1  *6+4    +1  *6+4    +1  *6+4    +1  *6+4  *6+2      =    =  ...
  143/104: [143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
  143/104: [143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
* 3 adjacent sequences
===Collatz roads===
124/109: [124 m  62 m  31 d  94 m  47 d 142 m  71 d 214 m 107 d 322 m 161 d 484 m  242 m 121 d | 364 ...
We define a "road" (with 2 parallel "lanes") as a sequence of pairs of elements (in 2 Collatz sequences with adjacent start values, read from right to left). A road is constructed by taking some n (364 in the example, the last common element of the 2 sequences) with n = 4 mod 6, and by applying the steps
            +1  *6+4  *6+2    =     =     =     =     =     =     =    =    =      =    =      = ...
  d m m d m d m d ...  
125/109: [125 d 376 m 188 m  94 m  47 d 142 m  71 d 214 m 107 d 322 m 161 d 484 m  242 m 121 d | 364 ...
  m m d m d m d m ...
            +1  +2/6    +2    +1  *6+4    +1  *6+4   +1  *6+4    +1  *6+4    +1  *6+4  *6+2      = ...
in alternating sequence, until one of the elements in the pairs becomes divisible by 3. The construction yields one road with an upper lane (left elements of the pairs) and a lower lane (right elements). In the first attempt one can construct the roads for all starting values 4, 10, 16, 22 ... 4+6*n, and list them as rows with lists of pairs:
  126/109: [126 m  63 d 190 m 95 d 286 m 143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m | 364 ...
* 4 adjacent sequences
314/38: [314 m 157 d 472 m 236 m 118 m  59 d 178 m 89 d 268 m 134 m  67 d 202 m 101 d 304 m 152 m 76 m 38 m
            +1  *6+4   +1  *6+4  *6+2  *6+1  *6-2  *6-1  *6-8  *6+4  *6-2    -2    -1  -4/6  -2/6
315/38:  [315 d 946 m 473 d 1420 m 710 m 355 d 1066 m 533 d 1600 m 800 m 400 m 200 m 100 m  50 m  25 d 76 m 38 m
            +1  -2/6  +1/6  +8/6  +4/6    +3  +8/6    +5  +16/6    +8    +4   +2    +1  *6+4  *6+2
316/38:  [316 m 158 m  79 d  238 m 119 d 358 m  179 d 538 m  269 d 808 m 404 m 202 m 101 d 304 m 152 m 76 m 38 m
            +1  *6+4  *6+2      =    =    =      *    *      =    =    =    =    =    =    =
317/38:  [317 d 952 m 476 d  238 m 119 d 358 m  179 d 538 m  269 d 808 m 404 m 202 m 101 d 304 m 152 m 76 m 38 m
 
780/122: [780 m  390 m  195 d  586 m  293 d  880 m  440 m  220 m  110 m  55 d 166 m  83 d 250 m 125 d 376 m 188 m 94
781/122: [781 d 2344 m 1172 m  586 m  293 d  880 m  440 m  220 m  110 m  55 d 166 m  83 d 250 m 125 d 376 m 188 m 94
782/122: [782 m  391 d 1174 m  587 d 1762 m  881 d 2644 m 1322 m  661 d 1984 m 992 m 496 m 248 m 124 m  62 m  31 d 94
783/122: [783 d 2350 d 1175 d 3526 m 1763 d 5290 m 2645 d 7936 m 3968 m 1984 m 992 m 496 m 248 m 124 m  62 m  31 d 94
 
* 5 adjacent sequences
86/121:  [386,  193, 580,  290, 145, 436, 218, 109, 328, 164, 82,  41, 124,  62, 31, 94, 47, 142, 71
87/121: [387, 1162, 581, 1744, 872, 436, 218, 109, 328, 164, 82,  41, 124,  62,  31, 94, 47, 142, 71
88/121:  [388,  194,  97,  292, 146,  73, 220, 110,  55, 166, 83, 250, 125, 376, 188, 94, 47, 142, 71
89/121:  [389, 1168, 584,  292, 146,  73, 220, 110,  55, 166, 83, 250, 125, 376, 188, 94, 47, 142, 71
90/121:  [390,  195, 586,  293, 880, 440, 220, 110,  55, 166, 83, 250, 125, 376, 188, 94, 47, 142, 71
 
418/41:  [418,  209, 628,  314, 157, 472, 236, 118,  59, 178,  89, 268, 134,  67, 202, 101,
419/41:  [419, 1258, 629, 1888, 944, 472, 236, 118,  59, 178,  89, 268, 134,  67, 202, 101,
420/41:  [420,  210, 105,  316, 158,  79, 238, 119, 358, 179, 538, 269, 808, 404, 202, 101,
421/41:  [421, 1264, 632,  316, 158,  79, 238, 119, 358, 179, 538, 269, 808, 404, 202, 101,
422/41:  [422,  211, 634,  317, 952, 476, 238, 119, 358, 179, 538, 269, 808, 404, 202, 101,

Revision as of 16:32, 24 August 2018

Introduction

When all Collatz sequences are read backwards, they form a graph starting with 1, 2 ..., hopefully without cycles (except for 1,2,4,1,2,4 ...). At each node n in the graph, the path starting at the root (4) and with the last node n can in principle be continued to 2 new nodes by a

  • "m"-step: n * 2 (which is always possible), or a
  • "d"-step: (n - 1) / 3 (which is possible only if n - 1 mod 3 = 0).

When n mod 3 = 0, the path will continue with m-steps only, since the duplication maintains the divisibility by 3.

Literature

Motivation: Patterns in sequences with same length

When Collatz sequences are investigated, there are a lot of pairs of adjacent start values with the same sequence length, and with a characteristical neighbourhood of every other value, for example (from OEIS A070165): 

142/104: [142 m  71 d 214 m 107 d 322 m 161 d 484 m  242 m 121 d | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]
           +1  *6+4    +1  *6+4    +1  *6+4    +1   *6+4  *6+2      =    =  ...
143/104: [143 d 430 m 215 d 646 m 323 d 970 m 485 d 1456 m 728 m | 364, 182, 91, ... 10, 5, 16, 8, 4, 2, 1]

Collatz roads

We define a "road" (with 2 parallel "lanes") as a sequence of pairs of elements (in 2 Collatz sequences with adjacent start values, read from right to left). A road is constructed by taking some n (364 in the example, the last common element of the 2 sequences) with n = 4 mod 6, and by applying the steps 

d m m d m d m d ... 
m m d m d m d m ...

in alternating sequence, until one of the elements in the pairs becomes divisible by 3. The construction yields one road with an upper lane (left elements of the pairs) and a lower lane (right elements). In the first attempt one can construct the roads for all starting values 4, 10, 16, 22 ... 4+6*n, and list them as rows with lists of pairs:

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