[seqfan] Re: Periodicity of seqs mod m - research idea

Joerg Arndt arndt at jjj.de
Sun Feb 19 09:20:54 CET 2017

```A word of warning:
With a (linear, homogeneous, constant coefficient) recurrence of order r
we do know that if r successive terms agree then we found a period
(see the code in https://oeis.org/A271953 ).

IIRC we do not even have to worry about a pre-periodic segment.

With A214551 (not: A215551 as said below) things seem to be much more
involved.  Could it not be that if we do observe some cycle mod m (for
some m), say, twice in a row, it does not repeat further up?

Looking at A214551 mod 2 strongly suggests that!

Best regards,   jj

* Neil Sloane <njasloane at gmail.com> [Feb 19. 2017 08:50]:
> Dear Seq Fans,  Here is an idea for research that I don't have time to work
> on myself, in case any one would like to play around with it.
>
> Fact: Fibonacci numbers (A45) are periodic mod m for any m. The periods (I
> mean period length, always) are called Pisono periods, A1175.
>
> The same is true for any linear recurrence with constant
> coefficients.
>
> Another example: Narayana's cows seq., A930, where the periods are given in
> A271901, A271953.
>
> Question: Are there any seqs defined by /nonlinear/ recurrences that are
> periodic mod m for some m (in a nontrivial way)?
>
> To test for the presence of a cycle (which in general won't start at the
> beginning of the sequence) the standard alg. is Floyd's hare and tortoise
> alg.  There is a Wikipedia article on Cycle Detection.
>
> I looked superficially at Recaman (A5132), EKG (A64413), Hofstadter
> (A5185), and a couple of other favorite recurrences but not find anything.
> This doesn't mean much, I didn't go very far and I only tried a couple of
> values of m.
>
> One promising candidate to look at is Reed Kelly's mysterious version of
> the Narayana sequence, A215551. Is this periodic mod m for any m?  Probably
> not, but if it was that would be exciting, so worth a try.
>
> Neil
>
> --
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```