(2+i)x+1 sequence

[kohmoto]

    (2+i)x+1 sequence is defined as follows


    a(n)=((2+i)*a(n-1)+1)/(1+i)^k , where (1+i)^k is the highest power of 
(1+i) dividing (2+i)*a(n-1)+1 .  -D1-
　
    If a(n-1) is not divided by  1+i , then (2+i)*a(n-1)+1  must be divided 
by  1+i .
    So, the following definition is equivalence as D1.

    a(n)=((2+i)/(1+i)*a(n-1)+1/(1+i))/(1+i)^k , where (1+i)^k is the highest 
power of (1+i) dividing (2+i)*a(n-1)+1 .  -D2-

    I think the coefficient  (2+i)/(1+i) is more essential than 2+i .
    And the norm is 10^(1/2)/2=1.581.... .

    [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.

    The known smallest record of A which make the sequence divergent  is 1.6 
in the case of p=2.
    Example : a(0)=1, A=1.6, B=1, p=2 . It seems to be unbounded.
　　　　
    If S3, T2 are unbound , then the norm is a record of A of Gaussian 
K-sequence.

    [Definition of GK-sequence]
    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 complex numbers. p is prime.
    a(n) is a Gaussian integer sequence.
    Where Floor[x+yi] defined as Floor[x]+Floor[y]*i

    Yasutoshi