Generalized GO

[koh]

----- Original Message ----- 
From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
To: "koh" <zbi74583 at boat.zero.ad.jp>
Sent: Wednesday, May 21, 2008 10:47 PM
Subject: Re: Generalized GO


> Dear Yasutoshi
> In my previous edits of similar sequences I suggested to change the %N
> to be shorter
> 
>>    %N A000001 Generate a sequence by the following rule.
>>               If b(n-1) is divisible by two then b(n) = b(n-1)/2.
>>               If b(n-1) isn't divisible by two then b(n) = k-(b(n-1)+1)/2.
>>               k is a integer.
>>               a(k) = {Number of cycles} * {The longest period} for each k
> 
> I think something like
> 
> a(k) = (Number of cycles)*(Longest period) of the sequence defined by
> b(n+1)=b(n)/2 if b(n) is even, =k-(b(n)+1)/2 else, b(0)=?.
> 
> would be much better.

    Thank you for editing.
    I will rewrite it.

    How about the following?

    If b(n-1)=0 Mod 2 then b(n)=b(n-1)/2
    If b(n-1)=1 Mod 2 then b(n)=k-Ceiling[b(n-1)/2]

   

> you did not give the initial term, which is important if not all
> numbers 1..k appear.
> U suppose "longest period" means length of longest cycle, and b(0)=k+1 (?)
> 

    I think both Number of cycles and Longest period don't depend on the initial term.
 
> OTOH I dont know why you multiply the 2 values (#cycles)(longest period)
> - why not keep these 2 more fundamental quantities in 2 seperate seq ?
> 

    I already posted  the sequence of number of cycles.

    %I A000002
    %S A000002 1,1,1,1,2,1,1,3,2,1,3,1    
    %N A000002 Generate a sequence by the following rule.
               If b(n-1) is    divisible by two then b(n) = b(n-1)/2.
               If b(n-1) isn't divisible by two then b(n) = k-(b(n-1)+1)/2.
               Sequence gives number of cycles for each k.              
    %e A000002 k=11
               11,5,8,4,2,1,10,5,....
               9,6,3,9,....
               7,7,....
               Three cycles exist. So, a(11)=3
    %Y A000002 A000001
    %K A000002 none
    %O A000002 1,5
    %A A000002 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp

    I have submitted it to OEIS.



    The A000001 was Generalized GO sequence b(n). I write it G.G.
    Neil said it didn't fit to OEIS.
    You know k and b(0) are arbitraly.
 
    I think an array of G.G. may fit.

    k    b(0)=1
    0    1 -1  0  0  0  0  0  0  0  0  0  0  ....
    1    1  0  0  0  0  0  0  0  0  0  0  0
    2    1  1  1  1  1  1  1  1  1  1  1  1
    3    1  2  1  2  1  2  1  2  1  2  1  2
    4    1  3  2  1  3  2  1  3  2  1  3  2  
    5    1  4  2  1  4  2  1  4  2  1  4  2
    6    1  5  3  4  2  1  5  3  4  2  1  5        
    7    1  6  3  5  4  2  1  6  3  5  4  2      
    8    1  7  4  2  1  7  4  2  1  7  4  2       
    9    1  8  4  2  1  8  4  2  1  8  4  2
    10   1  9  5  7  6  3  8  4  2  1  9  5 
    11   1  10 5  8  4  2  1  10 5  8  4  2     
    ....

    Main diagonal : 
         1,0,1,2,3,2,5,6,1,8,9,2

    The reason why I mulriplied the two is the following.

    Sometimes a(k) becomes k-1.
    I feel it is interesting.




    Yasutoshi