[seqfan] Re: The Fibonacci word over the nonneg integers

Neil Sloane njasloane at gmail.com
Sat Jul 1 14:57:16 CEST 2017

```Just to clean up this thread, A104234 has been edited, and Kerry's sequence

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Sat, Jul 1, 2017 at 7:00 AM, jean-paul allouche <
jean-paul.allouche at imj-prg.fr> wrote:

> Hi
>
> We should probably say: limit under iteration, starting from 0, of the
> morphism x --> x, x+1
> for even x and x --> x+1 for odd x , where the morphism is defined on the
> set of finite sequences
> of non-negative integers equipped with concatenation. Should we?
>
> best
> jp
>
> Le 30/06/17 à 23:55, David Seal a écrit :
>
>> As a minor wording point about:
>>
>> Trajectory of 0 under the map x -> x,x+1 if x is even, x -> x+1 if x is
>>> odd, starting at 0:
>>>
>>> 0, 1, 2, 2, 3, 2, 3, 4, 2, 3, 4, 4, 5, 2, 3, ...
>>>
>> I had a bit of trouble understanding what was intended - the map
>> described takes 0 to 0,1, so surely each 'point' on the trajectory contains
>> a 0, so why does the sequence contain only one 0 and not infinitely many?
>>
>> I have sorted out the answer to that question, but I think I would
>> describe the sequence as something like the limit of the trajectory rather
>> than the trajectory. I.e. the trajectory is the sequence of finite integer
>> sequences:
>>
>>     0
>> -> 0,1
>> -> 0,1,2
>> -> 0,1,2,2,3
>> -> 0,1,2,2,3,2,3,4
>> -> 0,1,2,2,3,2,3,4,2,3,4,4,5
>> -> 0,1,2,2,3,2,3,4,2,3,4,4,5,2,3,4,4,5,4,5,6
>> -> ...
>>
>> and the infinite sequence is in some sense what that trajectory converges
>> to.
>>
>> Looking at the finite sequences in that trajectory, and in particular how
>> many times each digit appeared in each, was entertaining - it gave me a
>> sort of 'diagonal sums' correspondence between the Fibonacci numbers and
>> Pascal's triangle that was new to me. Probably not actually new - they're
>> very well-studied mathematical constructs! - but quite fun:
>>
>>                   1 = 1
>>
>>                 1 + 1 = 3
>>                +
>>               1   2 + 1 = 8
>>                  +
>>             1 + 3   3 + 1 = 21
>>            +       +
>>           1   4 + 6   4 + 1 = 55
>>              +       +
>>         1 + 5   10+10
>>        +       +
>>       1   6 +15
>>          +
>>     1 + 7
>>    +
>>   1
>>
>> and at the alternate positions:
>>
>>                   1 = 2
>>                  +
>>                 1   1 = 5
>>                    +
>>               1 + 2   1 = 13
>>              +       +
>>             1   3 + 3   1 = 34
>>                +       +
>>           1 + 4   6 + 4
>>          +       +
>>         1   5 +10
>>            +
>>       1 + 6
>>      +
>>     1
>>
>> Best regards,
>>
>> David
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

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