[seqfan] Coordination sequences for planar nets
Neil Sloane
njasloane at gmail.com
Sun Nov 23 19:59:37 CET 2014
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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