Collatz Sequence Computed by a Tag System

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The Collatz sequence starts with a positive integer . If is even, then divide it by 2. If is odd, then multiply it by 3 and add 1. The sequence is the result of iterating this process.


We can modify this sequence by deleting all the numbers in the sequence by noticing that if is odd, then is even and the next number in the modified sequence should be .

Liesbeth De Mol described a set of rules for a tag system that computes this slightly modified view of the Collatz sequence. An -tag system is a machine that starts with a finite string of symbols, removes the first of them, and then appends some other string to the end, according to the value of the first symbol of the string at each click of the clock.

At each time step, the machine here removes two symbols from the beginning and then, depending on the first symbol, appends a string according to the rules:




The colors pink, red, and yellow stand for , , and , respectively.

Each number of the sequence (including the initial condition) is encoded as a string of successive "" symbols (or pink cells).


Contributed by: Emmanuel Garces Medina (March 2011)
Open content licensed under CC BY-NC-SA



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