[seqfan] Why A109794 is fascinating (and how you can help!)

Charles Greathouse charles.greathouse at case.edu
Mon May 2 05:10:34 CEST 2011

I often ask authors of new sequences to justify their importance.  This is
not a challenge, as such, but merely to encourage them to share what they
know or think is 'obvious' about the sequence.  Surely they think the
sequence is interesting or they would not have submitted it!

Perhaps sequence A109794 is such a case where the author thought its
qualities were self-evident, but there's not much to see at first glance.
It's a linear recurrence relation with the unenlightening definition "a(2n)
= A001906(n+1), a(2n+1) = A002878(n).".  It was submitted, presumably, as a
part of the author's program on "Floretions".  But it seems to have come up
in another context entirely, perhaps 'well-known to those who well-know it'.

It seems that members of this sequence are precisely those numbers n for
which the Pisano period of m is greater than the Pisano period of n for all
m > n.  That is, walking backward from infinity, these are the low-water
marks of A001175.

Can anyone prove the above assertion?  It may be easy, but I can't even
think of a good way to attack it at the moment.

This has practical implications: it means that the members of this sequence
are good candidates for checking whether a given number is a Fibonacci
number.  Take a member m of A109794 and compute the (small) set of Fibonacci
numbers mod m.  Then for any n, if n mod m is not in that set, n is not a
Fibonacci number.

Charles Greathouse
Case Western Reserve University

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