[seqfan] A sequence related to A030222 and perhaps others (?)

Thane Plambeck tplambeck at gmail.com
Sat May 20 21:23:12 CEST 2017

```Let Z2 be the usual set of integral points {x,y} in the Cartesian plane,
where x and y are both integers.

A high school student Michael Niszenson and I have been studying tilings of
the plane that are obtained by first deleting finitely many points from Z2,
and then computing the various polygonal Voronoi cell shapes C that arise
as each remaining *undeleted* point claims the Voronoi territory that is
nearest to it.

If amongst all the Voronoi cell shapes C that can arise in this way, a
particular polygonal tile shape (up to similarity) can be obtained by
deleting d points from Z2, but not d-1 (or fewer) deletions, then we're
saying that that polygon shape has birthday d.

So, for example, the (unit) square has birthday d=0.   At birthday one,
just one point of Z2 is missing, and the four nearest neighbors to it
divide up its territory amongst themselves, giving rise to four copies of
the unique birthday 1 tile, which has area 1.25.  We've christened that one
the "picket."

I hope this is somewhat clear.   I can send a paper with much better
pictures and intuition to anyone who would like to look at it.

Anyway, there are two related integer sequences we're interested in, one of
which we have some data for, and which seems to not be in the OEIS.   The
other one we've barely started on.

Sequence #1 One way to cast out a net so as to capture all conceivable tile
shapes born at birthday d is to enumerate deletion patterns of d points
that form a single connected graph component in the sense of being "at most
two units" away from one another.    This sequence seems to start 1, 3, 10,
68, and doesn't appear to be in the OEIS.  A030222, "n-polyplets," is
similar in spirit, with the "at most two units" in this paragraph, above,
replaced by "at most sqrt(2)" to get A030222.

https://oeis.org/search?q=A030222&sort=&language=english&go=Search

Sequence #2  This more natural sequence would count the number of
nonsimiliar shapes at each birthday.    We believe it starts 1, 1, 3, 6,
...    but it is not a program that we have good code for yet and the only
approach we have is to first have the values and shapes of sequence #1 in
hand to do the polygon "harvesting" if you will.

Anyway....a long message and probably there are mistakes.  But if people
would like to see our work so far send me a message and I'll send what we
have.

--
Thane Plambeck
tplambeck at gmail.com
http://counterwave.com/

```