[seqfan] Re: Coordination sequences for planar nets

W. Edwin Clark wclark at mail.usf.edu
Sun Nov 23 20:43:31 CET 2014


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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