A151514 -id:A151514 - cp's OEIS frontend

A182644 Number of fixed snake polyominoes with n cells.

1, 2, 6, 14, 34, 82, 198, 470, 1122, 2662, 6334, 14970, 35506, 83734, 198086, 466314, 1100818, 2587634, 6097830, 14316402, 33687146, 79008870, 185677006, 435098774, 1021404998, 2391646494, 5609151738, 13125214770, 30757286802, 71928506630

Offset: 1

Joseph Myers, Nov 24 2010

nonn

This sequence counts snake polyominoes both with and without holes; for example, it counts all four of these 7-cell snakes:
 ###  ##    ##  ###
 # #  # #  # #  # #
 ##   ###  ###   ##

Alain Goupil, Marie-Eve Pellerin, and Jérôme de Wouters d'Oplinter, Partially Directed Snake Polyominoes, arXiv:1307.8432 [math.CO], 2013-2014.

Snake polyominoes by group of symmetries relating shapes considered the same: A002013 (all symmetries), A182644 (translations only), A151514 (rotations and translations), A151527 (horizontal and vertical reflections, rotations of order 2 and translations), A151524 (reflections in either diagonal, rotations of order 2 and translations), A151523 (rotations of order 2 and translations), A151526 (reflections in a horizontal line and translations), A182646 (reflections in a NE-SW diagonal line and translations).

More terms from Alain Goupil and Jérôme de Wouters, Jun 21 2014

A151523 Number of 1-sided strip polyrhombs with n cells.

Original entry on oeis.org

1, 2, 4, 10, 20, 48, 106, 252, 578, 1372, 3208, 7584, 17850, 42102, 99276, 233716, 550960, 1295146, 3050226, 7161346, 16846668, 39511876, 92845830

Offset: 1

Ed Pegg Jr, May 13 2009

Also counts 1-sided strip polyrects.

Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyrhomb
Eric Weisstein's World of Mathematics, Polyrect

Strip polyominoes by group of symmetries relating shapes considered the same: A002013 (all symmetries), A182644 (translations only), A151514 (rotations and translations), A151527 (horizontal and vertical reflections, rotations of order 2 and translations), A151524 (reflections in either diagonal, rotations of order 2 and translations), A151523 (rotations of order 2 and translations), A151526 (reflections in a horizontal line and translations), A182646 (reflections in a NE-SW diagonal line and translations)

Edited and a(15)-a(23) by Joseph Myers, Nov 24 2010

A151526 Number of strip poly-IH64-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 4, 8, 19, 43, 103, 239, 569, 1339, 3185, 7503, 17794, 41908, 99137, 233251, 550624, 1294032, 3049412, 7158698, 16844717, 39505579, 92841149

Offset: 1

Ed Pegg Jr, May 13 2009

nonn

Equivalently, strip polyominoes where two polyominoes are considered the same if and only if they are related by a translation or a reflection in a horizontal line. Formerly described as one-sided strip polyrects, but that is A151523.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 6.2 and 9.4.

Strip polyominoes by group of symmetries relating shapes considered the same: A002013 (all symmetries), A182644 (translations only), A151514 (rotations and translations), A151527 (horizontal and vertical reflections, rotations of order 2 and translations), A151524 (reflections in either diagonal, rotations of order 2 and translations), A151523 (rotations of order 2 and translations), A151526 (reflections in a horizontal line and translations), A182646 (reflections in a NE-SW diagonal line and translations)

Edited and a(15)-a(23) by Joseph Myers, Nov 24 2010

A182646 Number of strip poly-IH68-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 7, 19, 41, 104, 235, 572, 1331, 3193, 7485, 17811, 41867, 99181, 233157, 550718, 1293817, 3049649, 7158201, 16845217, 39504435, 92842398

Offset: 1

Joseph Myers, Nov 24 2010

nonn

Equivalently, strip polyominoes where two polyominoes are considered the same if and only if they are related by a translation or a reflection in a NE-SW diagonal line.

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 6.2 and 9.4.

Strip polyominoes by group of symmetries relating shapes considered the same: A002013 (all symmetries), A182644 (translations only), A151514 (rotations and translations), A151527 (horizontal and vertical reflections, rotations of order 2 and translations), A151524 (reflections in either diagonal, rotations of order 2 and translations), A151523 (rotations of order 2 and translations), A151526 (reflections in a horizontal line and translations), A182646 (reflections in a NE-SW diagonal line and translations)

A359068 Number of 1-sided strip polyominoes with n cells.

Original entry on oeis.org

1, 1, 2, 5, 10, 24, 52, 124, 282, 668, 1548, 3654, 8533, 20093, 47033, 110533, 258807, 607227, 1421055, 3329585, 7785995, 18221563, 42575336, 99539106, 232398659, 542864111, 1266567155, 2956342341, 6893180336, 16078817198, 37469245219, 87347384305, 203447081205

Offset: 1

Arthur O'Dwyer, Jan 11 2023

A "strip" polyomino is a snake polyomino (A151514) with no holes.
This sequence first differs from A151514 at n = 7. An example of a polyomino counted by A151514, but not by this sequence, is:
  ###
  # #
  ##

Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi (gives first 33 terms).
Arthur O'Dwyer, C++ program.

A333313 gives the number of free (2-sided) strip polyominoes with n cells. Subtracting A333313 from A359068 gives the number of chiral pairs.
A151514 gives the number of 1-sided snake polyominoes with n cells; A151514(n) > A359068(n) for n >= 7.
Subtracting A359068 from A151514 gives the number of snake polyominoes with n cells that have at least one hole.

A359707 Number of 1-sided ouroboros polyominoes with k=2n cells.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 11, 45, 178, 762, 3309, 14725, 66323, 302342, 1391008, 6453950

Offset: 1

Arthur O'Dwyer, Jan 11 2023

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.

Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi (gives the first 32 terms)
Arthur O'Dwyer, C++ program

A151514 counts 1-sided snake polyominoes with k=n cells. A359707 added to A151514 gives the number of 1-sided polyominoes in which each cell has at most 2 (Von Neumann) neighbors.
A359706 counts free (2-sided) ouroboros polyominoes with k=2n cells. A359707 minus A359706 gives the count of chiral pairs. This sequence first differs from A359706 at k=14; the four chiral pairs of 14-cell ouroboroi are
    ###   ####   ###   ###
    # #   #  ##  # #   # ##
    # ##  ##  #  # ##  #  #
    #  #   ####  ## #  #  #
    ####          ###  ####
and their mirror-reflections.

cp's OEIS Frontend

A182644 Number of fixed snake polyominoes with n cells.

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A151523 Number of 1-sided strip polyrhombs with n cells.

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A151526 Number of strip poly-IH64-tiles (holes allowed) with n cells.

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A182646 Number of strip poly-IH68-tiles (holes allowed) with n cells.

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A359068 Number of 1-sided strip polyominoes with n cells.

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A359707 Number of 1-sided ouroboros polyominoes with k=2n cells.

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