[seqfan] Iteration

[Yasu Koh]

    Hi  Se fans

    Many years ago , I considered the following iteration
        x(n) = [A*x(n-1) + B]/p^k

        where [m] is Floor(m), p^k is the highest power of p dividing
[A*x(n-1) + B]

    It can strangely represent any degree sequence like Universal Turing
Machin  do  any computation
    I give you four example  that  re linear , two degree , three degree ,
four degree

    1.    Linear
        { x(0),p,A,B } = { 1,2,2.00013,3.0 } :
    Between 0-th term and 10000-th term, there are 44 linear subsequences
such that 4<={number of terms}.
The first several are
    x(16) to x(3613) x(n)=x(n-1)+2
    x(3650) to x(3655) x(n)=x(n-1)+1028
    x(3820) to x(3848) x(n)=x(n-1)+66
..........
>From a linear part to next linear part , it  behaves as if it were a random
sequence

    2.    Two degree
         { x(0), p, A, B } = { 2443499297,2,2.00014,3.0 } :
    The 68 terms from 0-th to 67-th are represented as follows
x(n) = 6*n^2+171040*n+2443499297

    3.    Three degree
        { x(0),p,A,B } = { 2417903893829791,2,2.00013,3.0 } :
     The 5 terms from 0-th term to 4-th term are represented as follows.
x(n) = 664/6*n^3+5107490*n^2+471475936498/3*n+2417903893829791

    4.    Four degree
        { x(0),p,A,B } = { 2015985557869547951,2,2.00013,3.2 } :
    The 8 terms from 0-th term to 7-th term are

2015985557869547951, 2016116596930809473, 2016247644509609977,
2016378700606503103, 2016509765222042527, 2016640838356781961,
2016771920011275153, 2016903010186075887

    I am sure polynomial sequence  of  five degree
    Could any one find it ?    It  might exist  on  LLI  {x0, p, A, B  } =
 {1, 2, 2.000014, 3.0

    Here  re more example

           https://mixi.jp/view_bbs.pl?comm_id=964735&id=99457040



    Yasutoshi