A324406 -id:A324406 - cp's OEIS frontend

A324407 Number of unoriented polyomino rings of length 4n with fourfold rotational symmetry.

1, 1, 1, 2, 3, 6, 10, 21, 38, 80, 157, 336, 691, 1493, 3164, 6900, 14880, 32647, 71212, 157069, 345216, 764666, 1689978, 3756879, 8338405, 18593389, 41410352, 92583361, 206790477, 463400376, 1037575558, 2329839141, 5227759707, 11759828568, 26436550400

Offset: 1

Robert A. Russell, Feb 26 2019

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 unoriented rings, a chiral ring and its congruent reflection are counted as one.
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

			For a(1)=1, the four cells form a square. For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted. For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted. For a(4)=2, the sixteen cells of one ring form a 5 X 5 square with the nine inner cells omitted; the other ring is similar, but with each corner cell omitted and replaced with the cell diagonally toward the center from that corner cell.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324406 (oriented), A324408 (chiral), A324409 (achiral).

Cf. A144553.

a(n) = A324406(n) - A324408(n) = (A324406(n) + A324409(n)) / 2 = A324408(n) + A324409(n).

A324408 Number of chiral pairs of polyomino rings of length 4n with fourfold rotational symmetry.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 12, 29, 61, 138, 294, 649, 1402, 3073, 6696, 14676, 32199, 70764, 156062, 344209, 762433, 1687745, 3751845, 8333371, 18582147, 41399110, 92557961, 206765077, 463343343, 1037518525, 2329710014, 5227630580, 11759537552, 26436259384

Offset: 1

Robert A. Russell, Feb 26 2019

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

			For a(5) = 1, the pair is   XXX          XXX .
                            X XXX      XXX X
                           XX   X      X   XX
                           X   XX      XX   X
                           XXX X        X XXX
                             XXX        XXX

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324406 (oriented), A324407 (unoriented), A324409 (achiral).

Cf. also A144553.

a(n) = A324406(n) - A324407(n) = (A324406(n) - A324409(n)) / 2 = A324407(n) - A324409(n).

A324409 Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 9, 9, 19, 19, 42, 42, 91, 91, 204, 204, 448, 448, 1007, 1007, 2233, 2233, 5034, 5034, 11242, 11242, 25400, 25400, 57033, 57033, 129127, 129127, 291016, 291016

Offset: 1

Robert A. Russell, Feb 26 2019

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.

			For a(1)=1, the four cells form a square.
For a(2)=1, the eight cells form a 3 X 3 square with the center cell omitted.
For a(3)=1, the twelve cells form a 4 X 4 square with the four inner cells omitted.
For a(4)=2, the sixteen cells of one ring form a 5 X 5 square with the nine inner cells omitted; the other ring is similar, but with each corner cell omitted and replaced with the cell diagonally toward the center from that corner cell.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

Cf. A324406 (oriented), A324407 (unoriented), A324408 (chiral).

Cf. A144553.

a(n) = 2*A324407(n) - A324406(n) = A324406(n) - 2*A324408(n) = A324407(n) - A324408(n).

A348848 Number of oriented polyominoes with 4n cells that have fourfold rotational symmetry centered at a vertex.

Original entry on oeis.org

1, 2, 6, 19, 65, 224, 790, 2851, 10424, 38496, 143454, 538667, 2035180, 7729146, 29486904, 112942373, 434114384, 1673766428, 6471199322, 25081542410, 97431694571, 379256586232, 1479022885116

Offset: 1

Robert A. Russell, Nov 01 2021

nonn

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.

			For a(1)=1, the polyomino is a 2 X 2 square. For a(2)=2, the two polyominoes are a chiral pair having a central 2 X 2 square with one cell attached to each edge of that square.

Robert A. Russell, Table of n, a(n) for n = 1..23
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A000988, A144553, A348849 (cell center).

Inner rings: A324406, A324407, A324408, A324409.

A348849 Number of fixed polyominoes with n cells that have fourfold rotational symmetry centered at the center of a cell.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 3, 6, 0, 0, 10, 18, 0, 0, 35, 57, 0, 0, 126, 191, 0, 0, 461, 658, 0, 0, 1699, 2308, 0, 0, 6315, 8241, 0, 0, 23686, 29853, 0, 0, 89432, 109268, 0, 0, 339473, 403450, 0, 0, 1294826, 1501074

Offset: 1

Robert A. Russell, Nov 01 2021

nonn

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.

			For a(9)=2, the polyomino is a 3 X 3 square or a row and column of five cells sharing their central cells.

Robert A. Russell, Table of n, a(n) for n = 1..96
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
Toshihiro Shirakawa, Enumeration of Polyominoes considering the symmetry, April 2012, pp. 3-4.

Cf. A000988, A144553, A348848 (vertex center).

Inner rings: A324406, A324407, A324408, A324409.

cp's OEIS Frontend

A324407 Number of unoriented polyomino rings of length 4n with fourfold rotational symmetry.

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A324408 Number of chiral pairs of polyomino rings of length 4n with fourfold rotational symmetry.

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A324409 Number of achiral polyomino rings of length 4n with fourfold rotational symmetry.

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A348848 Number of oriented polyominoes with 4n cells that have fourfold rotational symmetry centered at a vertex.

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A348849 Number of fixed polyominoes with n cells that have fourfold rotational symmetry centered at the center of a cell.

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