Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hello everyone. I am not a mathematician but recently I was thinking about one of Kimberling's questions he posted here: http://faculty.evansville.edu/ck6/integer/index.html , i.e the question I asked at the title. If we denote by a, b, c and d the number of appearances of the 4 basic strings '1', '11', '2', '22' in n-length sequence, then there would be an N-length sequence such that N = a + 2b + 2c + 4d [i] (we can consider the first, n-digits sequence as the longer one’s ‘run length’ sequence). Also, n = a + b + 2c + 2d [ii]. Now if indeed the proportion is ~ 50-50 between the 1’s and 2’s (with accuracy as high as we want), then for large enough n, I have noticed the following: When one tries to alter any one of a, b, c or d the ‘equilibrium’ is broken in a way that can’t be recovered: if, for instance, we make (a + 1) the number of ‘1’ strings in n. Then we must make (b – 1) the number of ‘2’ strings in order to keep [ii] but then in [i] we still have a problem. So let’s say we take (a + 2) to be the number of ‘1’ strings. Then we can take (c – 1) to keep both [i] and [ii] but on the other hand, that also means we have one less 2-letter string in the n-sequence, while we have 2 more 1-letter strings – but there is also a ‘run length’ sequence with n’ digits in which we have equal distribution of 1’s and 2’s – as we assumed – and that means the equilibrium in this sequence will be altered too, and so on… I noticed that any change of any of the 4 paramaters would cause similar disproportions. So to sum up, I don’t know how to formalize this well enough, and of course there’s a problem here – which is the fact that I already assumed the desired proportion and just ‘showed’ it can’t be altered. But, indeed, after checking with a computer program that for larger and larger n’s it’s about 50-50 between the 1’s and 2’s – it looks very clear to me that this is the answer. But can anyone help me prove it in a rigorous way? I just want to say again that as an undergraduate student I am sorry for lack of professionalism here but that’s the best I can do for now. And hope you will really understand my point…

share|improve this question
    
Does appear to be an open problem. Voting to close en.wikipedia.org/wiki/Kolakoski_sequence –  Will Jagy Feb 8 '12 at 4:13
1  
Working together to tackle unsolved mathematics problems is a great thing, but it's not what this website sets out to do - please see the faq. –  Gerry Myerson Feb 8 '12 at 4:16
    
One recent paper on this is cs.uwaterloo.ca/journals/JIS/VOL14/Bordelles/bordelles7r.pdf. Aside from the Wikipedia link Will Jagy provided, you might also want to check out oeis.org/A000002 which has numerous references as well as computer code. –  Joel Reyes Noche Feb 9 '12 at 0:37
add comment

closed as off topic by Will Jagy, Gerry Myerson, Andy Putman, Anthony Quas, Yemon Choi Feb 8 '12 at 5:23

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.