I stumbled upon [this](http://haroonsaeed.wordpress.com/2006/04/25/self-describing-numbers-and-sequence/) 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?



Self-describing number sequence
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.


