period

[y.kohmoto]

    Don Reble wrote,
>
>The first sequence has a period of 1790641. I had wondered whether it
>would ever repeat: the sequence grows unsteadily until the 477470'th
>term, when it reaches the value
>    82721263883011610886526961867779718135764506625125
>    11743699901796233367506991160435951413702199247578
>    48336819257023283527033772942121308102778129423389
>    10053098213870566201892389659879944244624498503906
>    76244208391103283037041703648964775995893652810637
>    75443503638514561304340346005699508846909678163541
>    30869180866795754559855095373101163156336097219855
>    71841626091479071631154016283655936808751927734552
>    38107621661269300592877592978724640932952936515280
>    23530580303113261797777433800708978953387458811266
>    82954756815787375433632658164067205860448170308260
>    27,
>a somewhat non-descript 552-digit number.
>

    Thank you very much. It is great.
    Which is the first term that the sequence goes into cyclic?
    What is the minimal in the  cycle?

>The second sequence is easy: it never goes past
>    81104929169215939436320658138839143546777420460293
>    79282485589496928568880735463557636136597534817627
>    54702800422427897905708391641024017523200893908310
>    4625
>and has a fundamental period of 180492.
>

    Is it correct?
    I couldn't verify it.
    At 411137th term, number becomes 370 digits.

     ----------

         Table of periods
         p=2, x(0)=107, 1.1<=a<=3.0,  0.5<=b<=1.4
       a  1.1           1.5              2.0                2.5
3.0
  b 0.5    1  1  1  2  1  3  1  1  1  1  1  -  89  -  -  -   -  2  -  -
    0.6    1  1  1  2  1  1  1  1  1  1  1  1  89  -  -  -   -  -  -  -
    0.7    1  1  1  2  1  1  1  1  1  1  1  1   -  -  -  -   -  -  -  -
    0.8    1  3  1  2  1  1  1  1  1  1  1  -   -  2  -  -   -  -  -  -
    0.9    1  3  2  2  1  1  1  1  1  1  4  -   4  2  -  -   -  -  -  -
    1.0    1  1  2  2  1  *  1  1  1  1  4  3   4  2  -  5   -  -  -  1
    1.1    5  1  2  2  1  -  1  1  1  1  4  3   2  2  -  5  60  -  -  1
    1.2    5  1  2  2  1  7  1  1  1  1  4  -   2  2  -  -  60  2  -  1
    1.3    1  1  2  2  1  7  1  1  1  1  3  -   -  2  -  -   1  2  -  1
    1.4    1  4  2  2  1  7  1  7  1  1  3  2   -  2  -  -   1  -  -  1
                          *=5083
         "-" means "probably divergent".
    A question :
    "Where is a  boundary of the area that K-sequences are divergent?"
    A candidate is sq1:{x(0), p, a, b = 107, 2, 1.6, 1.07}
    Another one is sq2:{x(0), p, a, b = 107, 2, 2.15, 0.5}
    If sq1 is not cyclic then the smallest value of the boundary is {a =
1.6}.
    I couldn't calculate the period of sq1, but I am not sure if it is
divergent.

    Yasutoshi
    http://boat.zero.ad.jp/~zbi74583/another02.htm