[seqfan] Coordination sequences (was "more")

[Neil Sloane]

Joseph, If I recall correctly, for some crystals there can be a problem.

Let L be a periodic structure with a specified set of edges joining the
points.
These edges might be bonds joining atoms, or they might be
lines joining points that are (say) a knight's move apart, or a bishop's
move apart.

The edges won't in general all have the same Euclidean length, certainly,
not even in chemical applications.

Let P be a point of L, and define the n-th crystal ball C_n(P) to consist
of all points
of L that are <= n steps away from P. The number of points in C_n is the
n-th partial sum of
the coordination sequence at P.

We run into problems if C_n is not convex, which does happen. There is an
extensive discussion here:

 *Low-Dimensional Lattices VII: Coordination Sequences
<http://neilsloane.com/doc/Me220.pdf>*, J. H. Conway and N. J. A.
Sloane, *Proc.
Royal Soc. London, Series A*, 453 (1997), pp. 2369-2389.
(there is a link to it in A008458).

There were also several follow-up papers by other people



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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