Generalized GO
koh
zbi74583 at boat.zero.ad.jp
Mon Jun 2 02:32:58 CEST 2008
----- 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
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