FAMP and recurrence relations

[Creighton Dement]

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