[seqfan] "reverse floretion lookup" Re: Shouldn't add links to recurrence relations without proof

mail at fumba.eu mail at fumba.eu
Mon Feb 15 21:12:57 CET 2010

Quoting Richard Mathar <mathar at strw.leidenuniv.nl>:

> To comment on  
> http://list.seqfan.eu/pipermail/seqfan/2009-December/003102.html :
> rm> The OEIS index on recurrence relations,
> rm> http://research.att.com/~njas/sequences/Sindx_Rea.html#recLCC does not
> rm> mention the Vincenzo Librandi sequences, perhaps because it is  
> the product
> rm> of manual effort, or some sort of periodic automatic process.
> The index has been created by digging manually through roughly the first
> 10 percent of the low A-numbers. (Corrections and additions are welcome!)
> I am detecting daily new sequences of that category, so the update  
> in the OEIS
> is done aperiodically.  I've added the current snapshot to my home page:
> http://www.strw.leidenuniv.nl/~mathar/progs/recLCC
> Richard Mathar www.strw.leidenuniv.nl/~mathar

I've added two index pages similar to this for 2nd order linear recurrences:

and 4th order linear recurrences (large data table > 2.5 MB):

which permit reverse lookups with "reasonable" ranges. This refers to  
starting with a linear recurrence relation of interest and finding the  
corresponding floretion(s) which generate recurrences of the same  
type. At the moment, only 1 floretion is given as an example for each  
recurrence and there are currently no direct links to OEIS sequences.  
Clicking on the link next to a recurrence should bring up the "Online  
Floretion Multiplier" loaded with a floretion which satisfies that  
recurrence. These pages are experimental and updated every few days- I  
haven't found all the ones I'm looking for yet!

On a side note, I noticed that if a floretion X generates sequences  
satisfying a recurrence of the form a(n+5) = A*a(n+4) + B*a(n+3) +  
C*a(n+2) + D*a(n+1),

then changing the sign of a floretion base vector (ieb(X), jeb(X),  
keb(X), eib(X), ejb(X), ekb(X), iib(X), jjb(X), kkb(X), ijb(X),  
ikb(X), jib(X), jkb(X)) has no effect on the coefficient B or the  
coefficient A and changing the sign of tes(X) has no effect on B but  
reverses the sign of A (these two statements follow immediately from  
"Floret's Equation").

Finally, to return to the original comment

the linear 4th order recurrences given for the floretions are not  
conjectures (they are proven by "Floret's Equation"). Of course, that  
does not exclude the possibility program bugs so one should always  
double check.


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