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

jean-paul allouche jean-paul.allouche at imj-prg.fr
Sat Jul 1 13:00:50 CEST 2017


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
>
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