Taming this Conway-type sequence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:03:42Z http://mathoverflow.net/feeds/question/34921 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34921/taming-this-conway-type-sequence Taming this Conway-type sequence To be cont'd 2010-08-08T12:59:58Z 2010-08-10T06:24:27Z <p>(I started working on this problem after trying to get any "interesting" pattern out of the number that Gowers randomly wrote while answering:<a href="http://mathoverflow.net/questions/34843/what-is-realistic-mathematics" rel="nofollow">What is realistic mathematics?</a>.)</p> <p>The number was 123871205412470874297947938271423698765734564756028492656.</p> <p>Take any number, for instance:</p> <blockquote> <p>123871205412470874297947938271423698765734564756028492656</p> <p>3484756955 (in the preceding number there are three 0's, four 1's,..., and finally five 9's)</p> <p>0001231111 (in the preceding number there are no 0's, no 1's,..., and finally one 9)</p> <p>3511000000</p> <p>6201010000</p> <p>.... clearly the list won't end as there will always be some no 0's or 1's or 2's, etc.</p> </blockquote> <p>A variant of this sequence is discussed <a href="http://mathoverflow.net/questions/27324/what-are-some-naturally-occurring-high-degree-polynomials" rel="nofollow">here(Conway's look-and-say sequence)</a>.</p> <p>This a fairly simple obsevation, so is there a literature about such sequences from which I can learn more?</p> http://mathoverflow.net/questions/34921/taming-this-conway-type-sequence/34929#34929 Answer by dke for Taming this Conway-type sequence dke 2010-08-08T14:42:02Z 2010-08-08T15:21:54Z <p>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 <a href="http://www.jstor.org/pss/2974579" rel="nofollow">an article</a> 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. </p> <p>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.</p>