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I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants
where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) for a few numbers.

It would be nice to support the article with some quality analysis.

Before June 9, 2011, was starting to extract and clearly define a pattern to these constants cf the article. I've made some progress in finding the patterns to their continued fractions. Can you find the next pattern $a_6(n)$? If it is like $a_5(n)$ it will be dependent on the moduli of some natural number $K$.

Beginning with $n=0$, the sequence is:
Edit [I made this to hard with an error. I should have said, Beginning with $n=1$, the sequence is:]

1, 2, 1, 3, 3, 9, 4, 1, 5, 2, 7, 9, 8, 1, 9, 2, 11, 2, 12, 1, 13, 2, 15, 16, 16, 1, 17, 2, 19, 1, 20, 1, 21, 2, 23, 1, 24, 1, 25, 2, 27, 1, 28, 1, 29, 2, 31, 1, 32, 1, 33, 2, 35, 2, 36, 1, 37, 2, 39, 2, 40, 1, 41, 2, 43, 2, 44, 1, 45, 2, 47, 2, 48, 1, 49, 2, 51, 3, 52, 1, 53, 2, 55, 3, 56, 1, 57, 2, 59, 3, 60, 1, 61, 2, 63, 3, 64, 1, 65, 2, 67, 4, 68 ,1, 69, 2, 71, 4, 72, 1, 73, 2, 75, 4, 76, 1, 77, 2, 79, 4, 80, 1, 81, 2, 83, 4, 84, 1, 85, 2, 87, 5, 88, 1, 89, 2, 91, 5, 92, 1, 93, 2, 95, 5, 96, 1, 97, 2, 99, 5, 100, 1, 101, 2, 103, 6, 104, 1, 105, 2, 107, 6, 108, 1, 109, 2, 111, 6, 112, 1, 113, 2, 115, 6, 116, 1, 117, 2, 119, 7, 120, 1, 121, 2, 123, 7, 124, 1, 125, 2, 127, 7, 128, 1, 129, 2, 131, 7, 132, 1.

The function for the pattern to the sequence could be piecewise defined, as the last known piece of $a_5(n)$ started at $n=24$. Whether the functions for $a_5(n)$ and $a_6(n)$ can be defined without piecewise functions has yet to be answered as far as I know.

Addendum June 10, 2011 [Now that I have $a_1(n)$ through $a_6(n)$, I find it difficult to find what they all have in common. Perhaps there is an easier pattern to find $a_1(n)$ from $a_0(n)$, $a_2(n)$ from $a_1(n)$, ...? Can you help me find it? I will ask something like this in the talk page soon.]

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    Apart from typos and the fact that any understandable explanation should use simple examples, you cannot use "truly random" without explanation and you cannot used "it is believed" instead of saying whether this is your personal conjecture or someone else's conjecture. I will vote to close, but I will certainly vote to reopen if you edit your question, explain this conjecture and ask us if it makes sense / has relations to known conjectures and results.2011-05-19
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    Marvin: (+1) I'd vote to reopen, but don't yet have the "threshold rep" to do so. Also @Zev: could you delete, or modify, your comments given the effort Marvin has put into modifying the question, and the sincerity of the question?2011-05-19
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    @Marvin: I've started a meta thread (http://meta.math.stackexchange.com/questions/2214/reopening-the-question-about-convergents) for gaining support for reopening the question.2011-05-20
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    @Amy J.M.: I will remove my original comments, because Marvin has edited his wiki page to improve most of the aspects I touched on. However, I still stand by my vote to close. [Myself's post](http://meta.math.stackexchange.com/questions/2214/reopening-the-question-about-convergents/2218#2218) on the meta thread pretty much captures my thoughts on this question.2011-05-20
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    @Zev: fair enough. I hope you didn't take my suggestion as an order? I didn't intend for my comment to come across that way. I just worry that new visitors, browsing questions for the first time (e.g. after they've been improved, etc) will read comments that don't necessarily apply anymore, and wonder "what the he...ck? Yikes!"2011-05-20
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    This reminds me of Khinchin's constant http://en.wikipedia.org/wiki/Khinchin's_constant because it is also the limit value of a function on _almost_ _all_ continued fraction parameters.2011-05-21
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    The link says "for most 10 < x < 11 the process returns 10.098057706624427..., not true for x = 10.1, 10.2, 10.5 and perhaps for other values." I wonder if it is true for sqrt(101)? According to the wiki article in my previous comment, solutions of quadratic equations do _not_ produce Khinchin's constant.2011-05-21
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    @Dan Brumleve, The input of sqrt(101) into the function of iterated continued fractions from convergents gives 10.0980577066244279660274026688, when you use the Mathematica code, l = Sqrt[101]; Table[c = Convergents[l, 100]; l = FromContinuedFraction[c], {n, 1, 50}]; N[l, Floor[30]] .2011-05-21
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    That value (which I guess is the same for most numbers between 10 and 11) to the precision it is given has three instances of "66" and three instances of three different other double digits ("77", "44", and "88"). I wonder why _that_ is true?2011-05-23
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    Dan, I don't know if we are looking too hard. The pattern might lie in the denominators of the simple continued fraction. See https://oeis.org/wiki/Talk:Table_of_convergents_constants#Partial_denominators_pattern .2011-05-23
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    I suggest that if you can find a way to extract this pattern (in base k) it may yield a proof that the attractors are irrational; I am thinking of the http://en.wikipedia.org/wiki/Proof_that_e_is_irrational2011-05-24
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    @Dan Brumleve, I'm working a little harder to extract patterns to the continued fractions of these constants.2011-05-30
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    @Dan: Re your numerological comment, note that even if the digits are selected at random, you expect to see a repeated digit at least once every $10$ digits (depending on the distribution of digits).2011-05-31
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    I'm not sure it makes sense to calculate the continued fraction of the constant. It is a fixed point of some operation on continued fractions with non-integral coefficients, so you shouldn't expect much from its integral continued fraction.2011-05-31
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    Yuval, indeed, it is numerological, but that is science. Marvin has now edited the question to render all comments out of context.2011-05-31
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    I take my last comment back, given the new interpretation of the iteration.2011-05-31
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    @Dan and etal, I'm sorry if my zeal for a more useful question made anyone's comment look out of place. I was afraid that if I started a new question about my work on that article, it would be closed as a duplicate, since this question was so general at first. P.S. It might take a while for me to digest all the comments and answers, but I think they will eventually help a lot!2011-06-01
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    Marvin, don't sweat it. If you have other questions or puzzles about continued fractions (even if you already know the answers) you would be doing the site a service by posting them; currently there are only 17 questions tagged as such out of more than 13000 total. I've found that posting questions is a good way to practice engaging the community and to understand what other people find interesting and why.2011-06-01

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