[seqfan] Periodicity of seqs mod m - research idea

Neil Sloane njasloane at gmail.com
Sat Feb 18 20:44:07 CET 2017


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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