Self-describing number sequence - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T16:33:26Z http://mathoverflow.net/feeds/question/24824 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/24824/self-describing-number-sequence Self-describing number sequence Ingdas 2010-05-15T22:00:16Z 2010-05-15T23:16:48Z <p>I stumbled upon <a href="http://haroonsaeed.wordpress.com/2006/04/25/self-describing-numbers-and-sequence/" rel="nofollow">this</a> number sequence while surfing the web. And I generated the next terms with my pc, and I was amazed to see, only the numbers 1 to 3 come up in megabytes of output. The sequence describes the previous number. The first term is one one, so 11 is the second one. The second consist 2 ones, so the third element is 21.</p> <pre><code>1 1 1 2 1 1 2 1 1 1 1 1 2 2 1 … … </code></pre> <p>So now I'm wondering, if there's any good explanation why only the numbers 1 to 3 come up? Or does any higher number comes up later on?</p> http://mathoverflow.net/questions/24824/self-describing-number-sequence/24826#24826 Answer by Quantumplation for Self-describing number sequence Quantumplation 2010-05-15T22:30:57Z 2010-05-15T22:30:57Z <p>If a 4 existed in the sequence, as such:</p> <p>... 41 ... </p> <p>That would mean that there were 4 one's in the previous sequence, as such:</p> <p>... 1111 ...</p> <p>So what does THAT describe? Basically it's saying "there's 1 occurrence of 1, followed by 1 occurrence of 1." Since you're describing consecutive numbers, that would ACTUALLY translate to "two occurrences of 1", or "21".</p> <p>Hope that helps.</p> http://mathoverflow.net/questions/24824/self-describing-number-sequence/24833#24833 Answer by Dan Piponi for Self-describing number sequence Dan Piponi 2010-05-15T23:04:51Z 2010-05-15T23:16:48Z <p>This sequence has been well studied by John Conway. Years ago I read his article in Eureka, the magazine of the mathematical society of Cambridge University, but I don't know if you can easily get hold of it now.</p> <p>You can find out a bit more <a href="http://en.wikipedia.org/wiki/Look-and-say_sequence" rel="nofollow">here</a>. In particular Conway shows that in the limit the sequence grows in length, on average, by a factor of $\lambda = 1.3035...$ at each step, with $\lambda$ a solution to a degree 71 polynomial. He also showed that there are essentially just 92 subsequences that repeat over and over again and never interact with each other. He (naturally) named these after chemical elements.</p>