[seqfan] Re: Rewriting squares

[Neil Sloane]

Yes, this slipped in from a different problem:  "Also the number of steps
to reach the limit cycle when starting from n," - Ignore it!


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.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com



On Wed, May 15, 2019 at 7:11 PM M. F. Hasler <seqfan at hasler.fr> wrote:

> 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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>