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Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. Each chiral ring is congruent to but different from its reflection; the two form a chiral pair.

These chiral rings have fourfold symmetry.

For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.

In early September, 2021, John Mason informed me that a(16) should be 6696 instead of 6695. He supplied me with representations of all of the rings, and I slowly discovered that my program had missed one and had serious errors. After I corrected it, we did match new values for a(16), a(18), a(20), and a(22). We are reasonably confident that the values shown are now correct. - Robert A. Russell, Sep 30 2021

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. Each achiral ring is identical to its reflection and has eightfold symmetry.

For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.

For k > 0, the numbers of achiral rings with 8k and 8k+4 cells are the same. In the former, there are four cells in the same row or column as the center tile; we obtain the latter by moving all the cells one-half a tile away from the center in both the horizontal and vertical directions, replacing those four center-line cells with four pairs of cells.

Redelmeier uses these rings to enumerate polyominoes of the regular tiling {4,4} with fourfold rotational symmetry (A144553) and an even number of cells. Each cell of a polyomino ring is adjacent to (shares an edge with) exactly two other cells. For oriented rings, chiral pairs (though congruent) are counted as two.

For n odd, the center of the ring is a vertex of the tiling; for n even, the center is the center of a tile.

Corrected; see A324408. - Robert A. Russell, Sep 30 2021

This sequence and its chiral and achiral versions correspond to Robert A. Russell's similar sequences for rings of fourfold rotational symmetry. The sequence does not count the mononimo or domino, referred to by Redelmeier as degenerate rings, as they are not in fact rings.

The sequence refers to rings with at least twofold (180-degree) rotational symmetry, and so includes those with (i) fourfold (90-degree) rotational symmetry, and (ii) all symmetries. - John Mason, Jan 19 2023

These are polyominoes of the regular tiling with Schläfli symbol {4,4}. For oriented polyominoes, chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the C90(n/4) sequence in the Shirakawa link. The calculation follows Redelmeier's method of inner rings.

These are polyominoes of the regular tiling with Schläfli symbol {4,4}. Chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the F90 sequence in the Shirakawa link. The calculation follows Redelmeier's method of determining inner rings.

A "snake" polyomino is a polyomino in which exactly two cells have exactly one (Von Neumann) neighbor apiece, and the rest have two neighbors apiece. Arthur O'Dwyer coined the term "ouroboros polyomino" for a polyomino in which every cell has exactly two neighbors: that is, an ouroboros polyomino is like a "snake" in which the head cell neighbors the tail cell.

A324407 etc. use the term "polyomino ring" in place of "ouroboros polyomino."

A checkerboard coloring shows that every ouroboros must have an even number of cells.

This sequence counts ouroboroi which do not designate a specific head or tail cell; thus the unique 8-cell ouroboros is

###

# #

###

One could imagine counting "headed" ouroboroi, in which the head and tail are distinguished. There are two distinct ways to create a free 8-cell "headed" ouroboros:

##H #HT

# T # #

### ###

This sequence first differs from A359707 (the count of 1-sided ouroboroi) at k=14. The four chiral 14-cell ouroboroi, each of which is counted once by A359706 and twice by A359707, are

### #### ### ###

# # # ## # # # ##

# ## ## # # ## # #

# # #### ## # # #

#### ### ####

A "snake" polyomino is a polyomino in which exactly two cells have exactly one (Von Neumann) neighbor apiece, and the rest have two neighbors apiece. Arthur O'Dwyer coined the term "ouroboros polyomino" for a polyomino in which every cell has exactly two neighbors: that is, an ouroboros polyomino is like a "snake" in which the head cell neighbors the tail cell.

A324407 etc. use the term "polyomino ring" in place of "ouroboros polyomino."

A checkerboard coloring shows that every ouroboros must have an even number of cells.