[seqfan] Re: A009927 (was Re: "more" keyword)

Neil Sloane njasloane at gmail.com
Sat Dec 19 01:56:21 CET 2015

Joseph, replying to that message

Maybe the only rigorous attack is to study the "skin" of the n-th crystal
ball, that is, the n-th shell.
If we can see that it is made up of A_n points of this type, B_n of this
type, etc., then we can
write down recurrences for this particular crystal, and then we are home

In other words, we analyze each crystal separately. That I will believe.

Best regards

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com

On Fri, Dec 18, 2015 at 6:20 PM, Joseph Myers <jsm at polyomino.org.uk> wrote:

> On Fri, 18 Dec 2015, Neil Sloane wrote:
> > > This does look like G.f.'s for other sequences of this type.
> >
> > Yes, but I'm not sure if any of them have been proved.  Looking back over
> > 20 years to this paper:
> >
> > R. W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic
> > Description of Coordination Sequences and Exact Topological Densities for
> > Zeolites <http://neilsloane.com/doc/ac96cs/>, Acta Cryst., A52 (1996),
> pp.
> > 879-889,
> >
> > I can't remember now if the g.f.s given there were empirical or if we
> > proved that they were correct. For a chemistry journal the distinction
> > didn't matter ....
> In Conclusions you say that "However, the results are empirical, as there
> is no rigorous mathematical proof that a generating function of the form
> (5) must hold for the CS of a periodic structure.".
> Proving the generating function is of that form is, I think, not too hard
> if you don't care about the proof corresponding to a practical algorithm.
> Getting reasonable bounds on the period lengths and when they start, so
> that the generating function can be deduced rigorously from the initial
> terms, is another matter, although I suspect it might actually be
> practical to get rigorous results for these sequences now (I'm pretty sure
> it should be practical for the 2-dimensional coordination sequences in
> OEIS).
> I note that a b-file has now been added for A009927.  It's not clear that
> b-file was based on the definition of the sequence rather than the
> empirical g.f.
> --
> Joseph S. Myers
> jsm at polyomino.org.uk
> _______________________________________________
> Seqfan Mailing list - http://list.seqfan.eu/

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