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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?
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    Dear Scott, I posted my question on meta as you asked me to do and you have not responded at all. Is there any reason for your silence?
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    "Dear Vassilis Parassidis, we have asked you to put your questions on meta.mathoverflow.net for approval before placing them on the main site. Please continue to do that." - S. Carnahan

    He said before, not afterwards.
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    Dear Scott, I will delete the question if that is will help. I want input on my question from others because the method is unique and original.
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    Dear Vassilis,

    please read the message I sent to you via email.

    Further, I think your comment above, including "Is there any reason for your silence?", addressed to the other Scott (Carnahan), is quite rude. As a moderator, he has no obligation to deal with your questions --- as a moderator, he's first trying to ensure the smooth running of the site.

    sincerely, Scott Morrison

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    For anyone else here: we had previously asked Vassilis to bring any question he was thinking about posting on MathOverflow to meta first, and only to post on the main site if he received a consensus that the question was appropriate. He ignored that request in this instance (only posting here after Scott C reminded him on this), and we've taken consequent action. (We're happy to explain this via private email if anyone is concerned or interested, but don't see an immediate need to explain here.)

    I'm going to leave this thread open, in case anyone is inclined to comment to Vassilis on the appropriateness of his question, or advise improvements.

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