[seqfan] 8/5 Sequence

kohmoto zbi74583.boat at orange.zero.jp
Mon Jun 11 11:01:40 CEST 2012

    Hi, Neil and Seqfan

    I think A082010 is not correct. it and the sequence of definition are different.
    If you started from 0 then it must be the following.

    It is not so interesting.
    The first interesting case is the following.



    Ask Don Reble.
    He has 2000000 terms of the sequence.

    Once Franklin wrote
    >This can be simplified: if x is odd, then next x is floor(8/5*x)+1.
    >No need for separate m and n; let r be the ratio (m/n), and if x is 
    >odd, the next x is floor(r*x)+1. And now we don't even need r to be 

    You are right.
    8/5 sequence is one of the K-Sequence.

    Where [N] is integer part of N, p^m is the highest p power dividing [A*a(n-1)+B]


    3 examples :
    See A028948, A029580, A036982

    >When r is the golden ratio, (sqrt(5)+1)/2, starting with 3, we get 
    >A001595, a sequence of all odd numbers, so there is unlimited growth. 
    >This may shed some light on what is happening with 8/5.

    I think your opinion is interesting.
    My observation is the following.

    p=2, a(0)=3, A=1.6, 1<=B<2
    In this area all sequences are unlimited. 

    You say the case of A=(1+root(5))/2 is also unlimited.
    How did you find this fact?
    I think it is your discovery. 

    Once I conjectured as follows.

    "K-Sequence which represents Fibonacci Sequence exists"

    I knew that A001959 is one of the K-Sequence.
    So my conjecture is almost right.


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