Self-describing number sequence I stumbled upon this 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.
1
1 1
2 1
1 2 1 1
1 1 1 2 2 1
…
…

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?
 A: 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.
You can find out a bit more here. 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.
A: If a 4 existed in the sequence, as such:
... 41 ... 
That would mean that there were 4 one's in the previous sequence, as such:
... 1111 ...
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".
Hope that helps.
