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