[seqfan] Re: Maximally symmetric "even" polyominoes

[John Mason]

On second thoughts, Robert A. Russell can certainly help, as he authored sequence A346800, which has a completely different definition but which apparently (needs proof) corresponds to your desired sequence.
(See values below)
Last year I formulated but didn’t prove the theory that (your proposed sequence) a(n) = 2*A006748(n) + 2*A056878(n) + A142886(n).
Where A006748 enumerates singly 45 degree reflectively symmetric polyominoes and A056878 enumerates those doubly so.
That said, the values you are looking for should be those below.
As you see, I have a different value for a(5) corresponding to size 20.
I find these (use Courier New):

__OO__
__OO__
OOOOOO
OOOOOO
__OO__
__OO__

OOOOOO
O____O
O____O
O____O
O____O
OOOOOO

_OOOO_
OO__OO
O____O
O____O
OO__OO
_OOOO_

__OO__
_OOOO_
OO__OO
OO__OO
_OOOO_
__OO__

_O__O_
OOOOOO
_O__O_
_O__O_
OOOOOO
_O__O_

That said, I think that it should be sufficient to check the above, and then introduce a comment in Robert’s sequence to say that it corresponds ALSO to what you were looking for.

john



1
0
2
1
5
4
16
13
54
46
186
167
660
612
2384
2267
8726
8464
32278
31822
120419
120338
452420
457320
1709845
1745438
6494848
6686929
24779026
25703792
94899470
99096382
364680344
383067646
1405619344
1484352159
5432421429
5764277096
21046198560
22429257682
81716371069
87432657722
317917129256
341394729018
1239120776640
1335080732960
4837744188806
5228480834780



From: Allan Wechsler
Sent: 27 January 2022 03:32
To: Sequence Fanatics Discussion list
Subject: [seqfan] Maximally symmetric "even" polyominoes

Some polyominoes have all the symmetry it is possible to have on a square
grid. The number of such maximally-symmetric polyominoes is counted by
https://oeis.org/A142886.

These polyominoes can be divided into two classes. First are the ones
typified by the monomino, which are centered on a cell. Then there are
those typified by the square tetromino, which aren't. I want to call these
"odd" and "even", respectively, but I fear this could be confusing, because
although the "even" class all have an even number of cells, the "odd" class
can have either an odd or an even number of cells. So maybe "face-centered"
and "vertex-centered"? Advice is welcome.

The "vertex-centered" examples all have order that is a multiple of 4. With
4 cells, the square tetromino is the lone example. With 8 cells, there are
no examples. With 12 cells, there are two different maximally-symmetric
dodecominoes. (For those who know how to decode these, they have encodings
F99F and 6FF6. Details on request.)

With 16 cells, there is just one example.

With 20 cells, there are 4; there are 2 with 24 and, if I haven't made a
stupid mistake, there are 10 with 28 cells.

You have probably guessed that I wouldn't be blathering on about this if
1,0,2,1,4,2,10... was in OEIS. You're right. It isn't. I think it should
be, but my polyomino-counting ability is poor. Does anybody have software
within easy striking distance of this census?

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