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Is this question acceptable?: Ratio of positive integers of a specific recursion

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    I was trying to express real zeros of the equation x^3+mx-1 as continued fractions and I came up with the following recursion method.
    a_1m^1+a_0=a_2, a_2m^2+a_1=a_3, a_n-1m^(n-1)+a_n-2=a_n
    k_1m^1+k_0=k_2, k_2m^2+k_1=k_3, k_n-1m^(n-1)+k_n-2=k_n.
    For m any non-zero positive integer and a_0=1, a_1=0, k_0=0, k_1=1 I was able to predict the result of k_n/a_n. For any other pair of values a_0≠a_1 and k_0≠k_1 the result is unpredictable. Does anyone know if it is possible to predict the numerical value of the ratio k_n/a_n for such pairs?
    If the question is not acceptable, how can I improve it?
    • CommentAuthorgrp
    • CommentTimeDec 23rd 2011
     
    It would help readers of meta if you followed a format like:
    <BEGIN FORMAT>
    Is the following question acceptable for posting on MathOverflow?

    <body of question>

    If it is not acceptable, how can it be improved?
    <END FORMAT>

    Your current question looks clear and understandable, except the formatting on meta makes it unclear to me whether
    it is two sets of three equations each (e.g X=Y, Z=W, V=a_n) , or two sets of multiple relations (e.g. X=Y=Z=a_n). To answer your question, you might consider drawing some kind of phase diagram for a system related to these recurrences like (x,y) goes (xm^i +a ,ym^i +b), and see if such a picture offers insight. Also, you might ask the question slightly differently, as in "Can one find the set of initial values (a_0,k_0,a_1,k_1) so that the sequence of values (k_n/a_n) converges, and can that limit of convergence be easily expressed in terms of the initial values?"

    Gerhard "Ask Me About System Design" Paseman, 2011.12.23
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    Dear Vassilis,

    Unless I am mistaken, this is a duplicate of your previous post on meta. Do you understand that I see this as spamming our site, and extremely unwelcome?

    Sincerely, Scott Morrison

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    Dear Scott,

    I thought that because I was suspended, members had considered the previous post closed. Because I am looking for answers I thought asking again was the right way to reopen the discussion. I certainly did not intend to spam the site.

    Sincerely,
    Vassili
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    Dear Gerhard,

    Thank you for your helpful comments. I will edit the formatting. I think "(e.g X=Y, Z=W, V=a_n)" would complicate the question

    Vassili
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    I still do not know if I am allowed to post my question. Can you please give me a definitive answer?

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