[seqfan] Polyomino miscellany

[Allan Wechsler]

Earlier I posted about my misadventures trying to write a Haskell program
to compute A048664, which counts oriented polyominoes with a particular
cell chosen as the "origin". Rotations, reflections, and different choices
of the origin cell are all considered different.

I had a bug in my program, which produced a sequence that grew quite a bit
faster than A048664 -- the first discrepancy was that a(3) was 24 instead
of 18.

I found the bug, and now know what I was counting, and the concept seems
interesting enough to add to OEIS, so I will probably do that in a few
days. For the moment I'm going to keep my explanation secret, in case
anyone wants to have fun figuring out the "secret rule" (knowing that it
arose from a buggy polyomino-counter).

(I fixed the bug, as well, and the program now produces A048664 just fine
-- except that it in excruciatingly slow.)

While fossicking among the polyomino sequences, I noticed A126202, which
also counts polyominoes with a single distinguished cell, but this sequence
regards rotations and reflections as equivalences. I was intrigued to note
that, unlike the extremely similar A048664, A126202 has an offset of 0; it
says there is one pointed nullomino.

Now, I am an enthusiastic supporter of the Zero Liberation Front. Zero is a
good number, the empty set is a a good set, and the nullomino is a good
polyomino, and should be counted wherever this can be remotely justified.
But I cannot see the justification for counting it (more than zero times)
in the case of A126202. The problem is that it has no cells from among
which to select the "origin". There is a nullomino, yes -- but there is no
nullomino with exactly one distinguished cell. In this case, I think
A048664 (which is 1-offset) got it right. If a(0) must be defined in either
case, I think, the only defensible value is 0, because there are no objects
satisfying the definition. Can anyone defend A126202(0) = 1?