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

[Neil Sloane]

OK, so here is a test case. Take the beautiful planar net 4.6.12, see
A072154 for the
coord. sequence, a nice pic from JJ, and a conjectured g.f. from Colin
Baxter.

Can the Seq Fans List come up with a proof that the conjectured g.f. is
correct?

Here there is only one kind of point.
Pick a starting point P.
Take colored pencils and start drawing the shells of points that are at
distance n from P.

After a while the successive shells will start to look kind-of periodic.

There will be shells of types (say) A B C A B C A B C ... and there should
be a recurrence for A_n in
terms of A_{n-1}, B_{n-1}, C_{n-1}, etc., and with luck we will end up by
showing the coordination
sequences is PORC ("polynomial on residue classes") and the g.f. will
hopefully
agree with what Colin found.

If we can do this case rigorously then the other empirical g.f.s will
be much more acceptable.

Of course this is only a 2-D example, we should really do a 3-D example too
(all the crystal structures like A009927 are 3-D, of course - and the
reason zeolites are so important is that they have cavities - making them
good catalysts - and so their crystal balls are not necessarily convex).



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 7:56 PM, Neil Sloane <njasloane at gmail.com> wrote:

> 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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>
>