[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
free.

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

Best regards
Neil

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
>
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