Generalized Go Sequence

[Peter Pein]

And even the length of the periods is in the EIS as
http://www.research.att.com/~njas/sequences/A003558.
I did not verify (prove) it, but the importance of powers of two is evident.

In[15]:=
goseq[k_] := Join[{1}, Most[NestWhileList[
     If[EvenQ[#1], #1/2, k - (#1 + 1)/2] & , k - 1, #1 =!= 1 & ]]]
In[16]:=
goseq[47]
Out[16]=
{1, 46, 23, 35, 29, 32, 16, 8, 4, 2}
In[20]:=
Length /@ goseq /@ Range[2, 100]
Out[20]=
{1, 2, 3, 3, 5, 6, 4, 4, 9, 6, 11, 10, 9, 14, 5, 5, 12, 18, 12, 10, 7,
12, 23,
  21, 8, 26, 20, 9, 29, 30, 6, 6, 33, 22, 35, 9, 20, 30, 39, 27, 41, 8, 28,
  11, 12, 10, 36, 24, 15, 50, 51, 12, 53, 18, 36, 14, 44, 12, 24, 55,
20, 50,
  7, 7, 65, 18, 36, 34, 69, 46, 60, 14, 42, 74, 15, 24, 20, 26, 52, 33, 81,
  20, 83, 78, 9, 86, 60, 29, 89, 90, 60, 18, 40, 18, 95, 48, 12, 98, 99}


the sequence of those k for which the mean of all values of a period of
goseq[k] (k>=2) is an integer value is:

1.) starting with
2,4,6,10,11,12,16,22,24,26,30,34,35,36,37,39,40,42,52,53,54,64,66,70,71,82,84,90,96,100,106,107,110,111,114,119,120,127,132,136,143,151

and 2.) _slightly_ too artificial for my taste ;-)

Peter

Joerg Arndt schrieb:
> What is the significance of 47?
> Also the seq appears to be trivially periodic(?)
> 
> What does the name refer to?
> 
> 
> * zak seidov <zakseidov at yahoo.com> [May 12. 2008 16:28]:
>> This is the same sequence
>> starting with 1:
>>
>>
> 1,46,23,35,29,32,16,8,4,2,
> 1,46,23,35,29,32,16,8,4,2,
> 1,46,23,35,29,32,16,8,4,2,
> 1,46,23,35,29,32,16,8,4,2,
> etc.
> 
>> and corresponding %N:
>>
>> %N A000001 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. b(0)=1. k=47.
>>
>> best, zak
>>
>> [...]
> 





Thanks for all the responses.  Mathematica 6.0 (32bit windows Feb 2008)
(run by Ray Chandler), DrScheme (programe by Joshua Zucker) and pari-2.3.3 (my
own Linux) agree on the results in these cases, whereas my own Maple 9
(Linux of 2003) differs from these by either +1 or -1.

Richard

Test cases for "number of unrestricted partitions p(n)" had been
n= 11269 11566 12376 12430 12700 12754 15013 17589 17797 18181 18421
18453 18549 18597 18885 18949 18997