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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?

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This is not too hard to prove, but these kinds of questions aren't really appropriate for this site; have you read the FAQ? There are suggestions for other websites to ask your questions. –  Qiaochu Yuan May 15 '10 at 22:18
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It is impossible to form a number in this sequence that has more than 3 identical digits in a row - just consider the possible cases. (Voting to close) –  S. Carnahan May 15 '10 at 22:24
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This is the Look and Say sequence of Conway. See en.wikipedia.org/wiki/Look-and-say_sequence for answers to your questions. –  Jason DeVito May 15 '10 at 23:02
    
I think that link to the OEIS should be added to faq. –  Nurdin Takenov May 15 '10 at 23:25
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closed as off topic by S. Carnahan, Qiaochu Yuan, Scott Morrison May 16 '10 at 19:33

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2 Answers

up vote 5 down vote accepted

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.

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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.

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