2.0001x+3 sequence

[kohmoto]

    Hello, Seqfans.
    Here is another kind of "Almost Integer ".

    [Definition of K-sequence] , K for Kimyo which means strange.
    a(n)=Floor[A*a(n-1)+B]/p^k , where p^k is the highest power of p 
dividing  Floor[A*a(n-1)+B] . A,B are real numbers. p is prime.

    If the coefficient  A is an "Almost Integer", more exactly, A=p^m+e , 
then the K-sequence becomes an "Extremsly Strange Sequence".
    Where e is a small positive real number.

    The sequences such that {A,p}={4+e,2}, {3+e,3}, {5+e, 5}, {7+e,7}... 
seem to belong in the same class of K-sequence.

    Only the sequene that {A,p}={2+e,2} is different from the other ESS.
    I named it a "LLI", for "a lot of linear parts which are isolated each 
other".

    It contains  also many polynomial sub sequences.
    All LLI might be periodic.

    I cojectured as follows.
         "limit  e --> 0  then {period of LLI}  --> infinite"

    Example :
    x0,p,a,b=1,2,2.00001,3.0         period=180492
    x0,p,a,b=1,2,2.00013,3.2         period=6366
    x0,p,a,b=1,2,2.00013,3.0         period=1790641
    x0,p,a,b=1,2,2.0001,3.0           period=13677
    x0,p,a,b=1,2,2.001,3.2             period=1670

    Yasutoshi