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

[Joseph Myers]

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