(I started working on this problem after trying to get any "interesting" pattern out of the number that Gowers randomly wrote while answering:[What is realistic mathematics?](https://mathoverflow.net/questions/34843/what-is-realistic-mathematics).)


The number was 123871205412470874297947938271423698765734564756028492656.


Take any number, for instance:



> 
> 123871205412470874297947938271423698765734564756028492656
> 
> 
> 3484756955 (in the preceding number there are three 0's, four 1's,..., and finally five 9's)
> 
> 
> 0001231111 (in the preceding number there are no 0's, no 1's,..., and finally one 9)
> 
> 
> 3511000000
> 
> 
> 6201010000
> 
> 
> .... clearly the list won't end as there will always be some no 0's or 1's or 2's, etc.
> 
> 
> 


A variant of this sequence is discussed [here(Conway's look-and-say sequence)](https://mathoverflow.net/questions/27324/what-are-some-naturally-occurring-high-degree-polynomials).


This a fairly simple obsevation, so is there a literature about such sequences from which I can learn more?



 Taming this Conway-type sequence
You have to be a bit more precise - for instance what happens if there are more than nine of a particular digit ? Regardless, a cursory literature search comes up with
[an article](http://www.jstor.org/pss/2974579) by Sauerberg and Shu which studies the Conway sequence as well as ones similar to yours, which are called factor-free counting sequences. The final section shows they are eventually periodic and gives a list of all possible cycles. 


In particular, when counting the digits 0,1,...,9 as in the question, one ends up either at the fixed point 6210001000 or at the 2-cycle 6300000100-->7101001000.


