[seqfan] Re: Rewriting squares

[jean-paul allouche]

Dear Maximilian

I think that the morphism in A308171
is not quite correct: the image of 9 should be 18
and the image of 8 should be 46. This is the same
morphism as for A308170. This morphism has
three infinite fixed points respectively beginning
with 1 (A308170), or with 5 (A308171) or with 6
(639181...). This last sequence might be worth adding.

best wishes
jean-paul



Le 16/05/2019 à 01:11, M. F. Hasler a écrit :
> On Sun, May 12, 2019 at 12:08 PM jean-paul allouche wrote:
>
>> By the definition itself, the infinite sequence is obtained by iterating a
>> morphism
>> (in the usual sens in combinatorics on words). For example, starting with 6
>> is exactly iterating the morphism
>> 6 --> 63
>> 3 --> 9
>> 9 --> 18
>> 8 --> 46
>> 1 --> 1
>> 4 --> 61
>> which gives 6 --> 63 --> 639 --> 63918 --> 63918146...
>>
> Indeed! In particular,
> the digit 7 (surprisingly chosen as initial value, rather than 3 or 9) will
> never occur.
> Similarly, when starting with 5 (A308171), we have to amend the above with
> 5 -> 52 ; 2 -> 4
> but the digits (5,2) = A308171(1..2) will never occur elsewhere again.
>
> Can it be proved or disproved that we can have A308170(n) = A308171(n+k)
> for some k and all n sufficiently large?
> What can be said / proved about the respective densities and /or positions
> - of the digits that occur infinitely often ?
> - where finite subsequence a(m..n), n>m>2, will occur again in the sequence?
>
> The sequence clearly is not squarefree (we have the cube 1, 6, 3, 1, 6, 3,
> 1, 6, 3 quite early)
> but can one make some other statement concerning squares, cubes... of given
> / minimal length ?
>
> On 12/05/2019 à 14:22, Neil Sloane wrote:
>>> It is certainly interesting.  We should have the two limiting sequences
>> in the OEIS:
>> 6,3,9,1,8,1,4,6,1,6,1,6,3,1,6,3,1,6,3,9,1,6,3,9,1,6,3,9,1,8,1,6,3,9,1,8,1,6,
>>> 3,9,1,8,1,4,6,1,6,3,9,1,8,1,4,6,1,6,3,9,1,8, ... ===> now  A308170
>>> or in the case of 25,
>>>
>> 5,2,4,6,1,6,3,1,6,3,9,1,6,3,9,1,8,1,6,3,9,1,8,1,4,6,1,6,3,9,1,8,1,4,6,1,6,1,
>>> 6,3,1,6,3,9,1,8,1,4,6,1,6,1,6,3,1,6,3,1,6,3, ... ===> now A308171
>
>>> Also the number of steps to reach the limit cycle when starting from n,
>
> It's unclear to me what could mean to "reach a limit cycle".
> Up to where the sequence(word) must coincide with the limit (which is never
> reached, except for the fixed point "1") ?
> Just the initial character? (This wouldn't be much interesting IMHO.)
> --
> Maximilian
>
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