Collatz Streetmap
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. MBK78
- File of first 10K Collatz sequences, ascending start values, with lengths
- The Collatz-Problem. A view into some 3x+1-trees and a new fractal graphic representation. Gottfried Helms, Univ. Kassel
- Collatzfolgen und Schachbrett, on Wikibooks
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: