2nd floretion paper / + Pell numbers

Creighton Dement crowdog at crowdog.de
Sat Sep 10 11:08:35 CEST 2005

```Good day to all,

My apologies... it appears the proof of the 2nd proposition cannot be
shown in the same way for n=3- not sure if the proposition itself is
simply incorrect or if it needs the help of some lemma. In any case, I
see determining the set G of all floretions x such that tesseq[x] is an
integer sequences (presumably all of 4th order or less) as an
interesting pursuit.

On a side note, here are two sequences satisfying 8th order recurrence
relations which sdd to the Pell numbers (disregarding signs):

2lesseq: -3, 4, 10, 41, 99, -140, -338, -1393, -3363, 4756, 11482,
47321, 114243, -161564, -390050, -1607521, -3880899, 5488420, 13250218,
54608393, 131836323, -186444716, -450117362, -1855077841, -4478554083,
6333631924,
G.f.
(-3+4*x+x^7+2*x^6-4*x^5-3*x^4+41*x^3+10*x^2)/((x^4-4*x^3+8*x^2-4*x+1)*(x^4+4*x^3+8*x^2+4*x+1))

and 2tesseq: -1, 0, 0, 17, 41, 0, 0, -577, -1393, 0, 0, 19601, 47321, 0,
0, -665857, -1607521, 0, 0, 22619537, 54608393, 0, 0, -768398401,
-1855077841, 0, 0, 26102926097, 63018038201, 0, 0
G.f.
(-1+x^7+7*x^4+17*x^3)/((x^4-4*x^3+8*x^2-4*x+1)*(x^4+4*x^3+8*x^2+4*x+1))

1lestesseq (= (2tesseq+2lesseq)/2 ): -1, 2, 5, 12, 29, -70, -169, -408,
-985, 2378, 5741, 13860, 33461, -80782
G.f.
(-1+2*x-5*x^4+12*x^3+5*x^2-2*x^5+x^6)/((x^4+4*x^3+8*x^2+4*x+1)*(x^4-4*x^3+8*x^2-4*x+1))

The slight problem is that the new FAMP Code which generates the
sequence needs to be tested for accuracy before I submit ( unless I
submit without the code, of course).

Sincerely,
Creighton

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> Date: Thu,  8 Sep 2005 15:35:38 +0200
> Subject: 2nd floretion paper
> From: "Creighton Dement" <crowdog at crowdog.de>
> To: seqfan at ext.jussieu.fr

> Dear Seqfans,
>
> I'm have begun a 2nd floretion paper- the main topic here is going to
> be integer sequences (there is still much too add- see some of the
> ideas floating around on my homepage, for example). Perhaps some of
> you would enjoy proof-reading the first 3 pages under
> http://www.crowdog.de/Flointseq.pdf
>
> The last proof is currently incomplete.  I've shown it for n=1 and
> n=2.
> I think it shouldn't  be too hard to finish the proof off with
> induction, but would like to know if what I've written is
> understandable up to that point.
>
> Finally, I need to find the (16) matrices which give an isomorph
> irreducible representation of the floretion basis vectors (a one-dim.
> representation has already been found- see first paper) . That could
> help with several proofs and should also ease tests for invertibility.
> It's somewhat embarrassing for me to admit that I hadn't set out to
> find them until now because of all the other stuff it appeared I
> needed to do first.  If you find them before I, I will definitely
> mention your name in the paper!
>
> Sincerely,
> Creighton
>
>
>
>

```