Number factorization and a highly suspicous claim

[Creighton Dement]

Dear Seqfans, 

Thanks very much to both Gottfried Helms and Max Alekseyev for their
informative replies as well as to Jonathan Vos Post for his initial
question. 

I wrote: 
 

> For x = -2'i -'j -'k,  the sequence
> (tes[(Fx+Gx)], tes[(Fx+Gx)^2], tes[(Fx+Gx)^3], tes[(Fx+Gx)^4], )
> "looks a lot like" two times the sequence
> http://www.research.att.com/projects/OEIS?Anum=A001333
> Numerators of continued fraction convergents to sqrt(2)
> 
> and by proposition 2.0.3 (Pure Quaternions and 2nd Order Recurrence
> Relations) 
>  
> 
> ********
> tes[(Fx+Gx)^n] = -x_0tes[(Fx+Gx)^(n-1)] - x_1x_2tes[(Fx+Gx)^(n-2)] for
> x = x_0'i + x_1'j + x_k'k
> ********
> the above sequence "really is" two times the OEIS sequence A001333
> (though a proof of the last proposition currently incomplete - see my
> SOS at the top of the page).


I believe the proof of 2.0.3  
http://www.crowdog.de/Flointseq.pdf

is now complete (though a proof of the previous lemma has not yet been
entered, it now only appears to be a formality.  This means that for any
 x of the form x = x_0 'i + x_1 'j + x_2 'k  and E = F + G = (1/4)( 'i +
i' + 'ii' + 'jj' + 'kk' + 'jk' + 'kj' + e), the sequences tesseq[Ex],
vesseq[Ex], lesseq[Ex], jesseq[Ex], etc.  all satisfy a single 2nd order
linear recurrence relation. It also means it is unnecessary, for
example, to use the generating functions of the various vesseq[Ex],
vesseq[Ex], lesseq[Ex], jesseq[Ex] to prove, for ex.:

vesseq[Ex] =   jesseq[Ex] + lesseq[Ex] + tesseq[Ex]
 
or other relations such as 

jesright[Ex] + jesleft[Ex] = jes[Ex]. 

It is trivial to show that for any floretion y:  vesseq[y] = jesseq[y] +
lesseq[y] + tesseq[y].  As noted before, the hard part seems to have
been to prove that vesseq[Ex], etc. actually satisfied a certain
recurrence relation. 


Thanks, 
Creighton