Floretion power sequences

[Creighton Dement]

Dear Seqfans, 

In a previous post (see below) I mention that, without a minimum amount
of theory to back it up, I see no direct way to conclude, for ex., that
a floretion sequence given by FAMP as (1, 2, 3, 4, 5, 6, 7...)  is the
sequence of naturals - regardless of how many terms are calculated.  

There now appears to be a proof that if F = .25('ii' + 'jj' + 'kk' + e)
and x is any floretion, then tesseq[Fx] really is a power sequence (as
opposed to a sequence that just looks a lot like it might be one). 

The results are given under: 
http://www.crowdog.de/Flointseq.pdf


My goal over the past few weeks has been to show that if E = .25('i + 'i
+ 'ii' + 'jj' + 'kk' + 'jk' + 'kj' +  e) and x is any floretion, then
tesseq[Ex] satisfies a second order linear recurrence relation with
constant coefficients. Note that E = .25('i + i' + 'jk' + 'kj) + F  -
hopefully it will be possible to use the above results in one way or
another.  If you have any ideas for me (including those on how to
simplify the notation given as much as possible), I would enjoy hearing
them.


Sincerely, 
Creighton 

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> Date: Tue,  9 Aug 2005 23:18:30 +0200
> Subject: Re: FAMP and recurrence relations
> From: "Creighton Dement" <crowdog at crowdog.de>
> To: seqfan at ext.jussieu.fr

> I wrote:
> 
> > 
> > The identity fam + fam* = ves holds, thus
> > fam + tes + fam* = famtes + fam* = ves + tes. So,
> > (1, -2, -5, 7, 19, -26, -71, 97,) + (-1, 0, 1, -1, -3, 4, 11, -15,
> > -41,) =
> > (0, 0, -4, -1, 16, 4, -60, -15,) + (0, -2, 0, 7, 0, -26, 0, 97, 0,
> > -362)
> > 
> > I'm currenctly trying to document the g.f.'s associated with various
> > floretions on my site http://www.crowdog.de/20801/46618.html
> > (note: the page is still under construction and the top link has not
> > been filled)
> > 
> 
> Here's something I consider important. I can take various floretions
> and document their g.f.'s forever... that doesn't change the fact,
> for. ex., that when FAMP returns
> tesseq[X]: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
> and I write down that the above sequence has g.f. x/(x-1)^2 - no proof
> at all has been given that what I'm seeing on the screen is actually
> the sequence of natural numbers.  It is just "assumed".
> 
> On the other hand, if the converse of the previously posted conjecture
> III given at http://www.crowdog.de/20801/46618.html is proved: If X is
> any floretion, then tesseq[X] satisfies a 4th order (or less)
> recurrence relation we are in good shape. Then, of course, one could
> look at tesseq[X]
> and say "I know it must satisfy a 4th order recurrence relation or
> less and I know it starts out 0, 1, 2, 3, 4, 5, 6, ...   Thus, it must
> have g.f. x/(x-1)^2".
> 
> This is potentially important. An ex.: Max recently gave a proof of a
> conjecture I made involving Pell numbers. That conjecture was actually
> a FAMP identity (i.e. the proof of the identity's validity is trivial-
> see the “Listen and Speak” article given on my homepage). If I would
> have been in a position to show that the floretion-generated sequences
> really corresponded to the g.f.'s given, there would have been nothing
> to prove (this is just my opinion- please let me know if I'm
> mistaken).  The relation given at the top of this page is a 2nd
> example.
> 
> 
> Many thanks,
> Creighotn
> 
> 
> 
> 
>