# [seqfan] Re: 8/5 Sequence

Maximilian Hasler maximilian.hasler at gmail.com
Tue Jun 12 19:19:51 CEST 2012

```AFAICS Yasutoshi thought that A082010 should be the orbit of 1 (with a
prefixed 0),
but of course it is the 8/5 *map* itself, not the orbit of an element.

(PARI)
A082010(x)=if(bittest(x,0),8*x\5+1,x\2)

vector(99,i,t=if(i>1,A082010(t),1))
%2 = [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1,...]

vector(99,i,t=if(i>1,A082010(t),3))
%3 = [3, 5, 9, 15, 25, 41, 66, 33, 53, 85, 137, 220, 110, 55, 89, ...]

Maximilian

On Tue, Jun 12, 2012 at 12:50 PM, Neil Sloane <njasloane at gmail.com> wrote:
> Yasutoshi, Thank you for posting that message.
> You say there is something wrong with A082010.
> The sequence looks correct to me, and so does
> the related sequence A152199.
> What exactly needs to be changed?
> Neil
>
> On Mon, Jun 11, 2012 at 5:01 AM, kohmoto <zbi74583.boat at orange.zero.jp>wrote:
>
>>    Hi, Neil and Seqfan
>>
>>
>>    I think A082010 is not correct. it and the sequence of definition are
>> different.
>>    If you started from 0 then it must be the following.
>>    0,1,2,1,2,....
>>
>>    It is not so interesting.
>>    The first interesting case is the following.
>>
>>
>>  3,5,9,15,25,41,66,33,53,85,137,220,110,55,89,143,229,367,588,294,147,236,118,59,95,153,245,393
>>
>>    http://list.seqfan.eu/pipermail/seqfan/2009-September/009818.html
>>
>>    He has 2000000 terms of the sequence.
>>
>>    Once Franklin wrote
>>    >This can be simplified: if x is odd, then next x is floor(8/5*x)+1.
>>    >No need for separate m and n; let r be the ratio (m/n), and if x is
>>    >odd, the next x is floor(r*x)+1. And now we don't even need r to be
>>    >rational.
>>
>>
>>    You are right.
>>    8/5 sequence is one of the K-Sequence.
>>
>>    a(n)=[A*a(n-1)+B]/p^m
>>    Where [N] is integer part of N, p^m is the highest p power dividing
>> [A*a(n-1)+B]
>>
>>
>>    3 examples :
>>    See A028948, A029580, A036982
>>
>>    >When r is the golden ratio, (sqrt(5)+1)/2, starting with 3, we get
>>    >A001595, a sequence of all odd numbers, so there is unlimited growth.
>>    >This may shed some light on what is happening with 8/5.
>>
>>    I think your opinion is interesting.
>>    My observation is the following.
>>
>>    p=2, a(0)=3, A=1.6, 1<=B<2
>>    In this area all sequences are unlimited.
>>
>>    You say the case of A=(1+root(5))/2 is also unlimited.
>>    How did you find this fact?
>>    I think it is your discovery.
>>
>>    Once I conjectured as follows.
>>
>>    "K-Sequence which represents Fibonacci Sequence exists"
>>
>>    I knew that A001959 is one of the K-Sequence.
>>    So my conjecture is almost right.
>>
>>
>>
>>    Yasutoshi
>>
>>
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>>
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>>
>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA