[seqfan] Re: Coordination sequences for planar nets

[Neil Sloane]

Edwin, See Gruenbaum and Shepherd's classic book, Patterns and Tilings.

There is also a wikipedia page, although I don't trust anything on that
site:  http://en.wikipedia.org/wiki/List_of_convex_uniform_tilings

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 Sun, Nov 23, 2014 at 2:43 PM, W. Edwin Clark <wclark at mail.usf.edu> wrote:

> Neil,
>
> What is the definition of a planar net? I'm sure not this:
>
>
> http://elementarymath.files.wordpress.com/2010/04/elementary-math-chapter-47.pdf
>
> --Edwin
>
> On Sun, Nov 23, 2014 at 1:59 PM, Neil Sloane <njasloane at gmail.com> wrote:
>
> > There are 11 uniform (or Archimedean) tilings in the plane.
> > If we take the 3^6 tiling (or net) (6 triangles around each point),
> > start at a lattice point, and walk outwards for 0, 1, 2, 3, 4, ... steps,
> > the number of points we reach for the first time gives the sequence 1, 6,
> > 12, 18, 24, 30, 36, 42, ...,
> > increasing by 6 at each step after the first.
> > This is sequence https://oeis.org/A008458 in the OEIS.
> > (In other words, it is the number of nodes at graph distance n from a
> fixed
> > node.)
> >
> > The planar net 3.6.3.6 gives https://oeis.org/A008579, and I just added
> a
> > primitive drawing to the entry to illustrate the first few terms. This is
> > rather more complicated.
> >
> > Next I looked at the 3^4.6 net, and for the initial terms of
> > the sequence I get 1,5,9,15,19,24, by hand.
> > This is bothersome, because (a) it is quite irregular, and (b) it was not
> > in the OEIS!  I just added it (https://oeis.org/A250120), along with a
> > drawing showing my calculations. I have no confidence in these numbers -
> > could someone check them?
> >
> > I don't know how many of the other planar nets are in the OEIS. 3^6 is
> > A008458, 3^4.6 is tentatively A250120, 3^3.4^2 is A008706, 3^2.4.3.4 = ?,
> > 4^4 is A008574, 3.4.6.4 is ?, 3.6.3.6 is A008579, 4.8^2 is A008576, 6^3
> is
> > A008486, and the others I don't know.
> >
> > Best regards
> > Neil
> >
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> >
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> >
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