[seqfan] Coordination sequences (was "more")
njasloane at gmail.com
Sat Dec 19 01:25:12 CET 2015
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
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
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.
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
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),
> > 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
> 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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