A002013 -id:A002013 - cp's OEIS frontend

A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

1, 1, 2, 4, 11, 27, 83, 255, 847, 2829, 9734, 33724, 118245, 416816, 1478602, 5267171, 18840144, 67611472, 243378415, 878407170, 3178068821, 11523323634, 41865833602, 152382134767

Offset: 1

Tanya Khovanova, Jul 27 2007

2n+2 is the maximal perimeter of an n-celled polyomino. a(n) is the number of n-celled polyominoes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge).

Andrew Clarke, Isoperimetrical Polyominoes, The Poly Pages.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).

Cf. A000105, A057730. Diagonal of A342243.
A359522 counts only polyominoes with holes.
A002013 counts only unbranched polyominoes.
A038142 is the analog for polyhexes.

a(n) <= A000105(n), a(n) <= A057730(n+1).

a(n) >= A000602(n) [see comment on edge graph trees]. - R. J. Mathar, Mar 08 2021

a(14)-a(16) from David Radcliffe, Dec 25 2017
a(17) from David Radcliffe, Dec 26 2017
a(18)-a(24) from John Mason, Dec 11 2021

A151514 Number of 1-sided snake polyominoes with n cells.

Original entry on oeis.org

1, 1, 2, 5, 10, 24, 53, 126, 289, 686, 1604, 3792, 8925, 21051, 49638, 116858, 275480, 647573, 1525113, 3580673, 8423334, 19755938, 46422915, 108783480, 255359883, 597932342, 1402308318, 3281352516, 7689369625, 17982241557, 42108302007, 98422076879, 230322745835

Offset: 1

Ed Pegg Jr, May 13 2009

A snake polyomino has 2 cells with 1 neighbor each, and (n-2) cells with 2 neighbors each. - Arthur O'Dwyer, Dec 12 2022

Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi, Dec 10 2022.
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyomino

A002013 counts 2-sided (free) snake polyominoes.

A359068 gives the number of 1-sided strip polyominoes (that is, snakes without holes) with n cells; A359068(n) < A151514(n) for n >= 7.

a(15)-a(23) from Joseph Myers, Nov 22 2010
a(24)-a(29) from Arthur O'Dwyer, Dec 10 2022
a(30)-a(33) from Arthur O'Dwyer, Jan 19 2023

A182644 Number of fixed snake polyominoes with n cells.

Original entry on oeis.org

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)

A333313 Number of 2-sided strip polyominoes with n cells and no holes.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 13, 30, 64, 150, 338, 794, 1836, 4313, 10067, 23621, 55313, 129647, 303720, 711078, 1665037, 3894282, 9111343, 21290577, 49770844, 116206114, 271435025, 633298969, 1478178004, 3446626028, 8039424324, 18734704111, 43673728357, 101723730306

Offset: 0

Peter Kagey, Mar 14 2020

nonn

This sequence first differs from A002013 at n = 7. An example of a polyomino counted by A002013, but not by this sequence, is:
  ###
  # #
  ##

Code Golf Stack Exchange, Counting polystrips
Arthur O'Dwyer, Polyomino strips, snakes, and ouroboroi, 2022.

Cf. A002013.

a(16)-a(26) from John Mason, Sep 15 2022

a(27)-a(33) from Arthur O'Dwyer's blog added by Andrey Zabolotskiy, Jun 20 2024

A359706 Number of free (2-sided) ouroboros polyominoes with k=2n cells.

Original entry on oeis.org

0, 1, 0, 1, 1, 4, 7, 31, 95, 420, 1682, 7544, 33288, 152022, 696096, 3231001

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.
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
    ###   ####   ###   ###
    # #   #  ##  # #   # ##
    # ##  ##  #  # ##  #  #
    #  #   ####  ## #  #  #
    ####          ###  ####

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

A002013 counts free (2-sided) snake polyominoes with k=n cells. A359706 added to A002013 gives the number of free polyominoes in which each cell has at most 2 (Von Neumann) neighbors.

A359707 counts free (2-sided) ouroboros polyominoes with k=2n cells. A359706 subtracted from A359707 gives the count of chiral pairs.

A359521 Number of free mature snake polyominoes of n cells, where mature means that the snake cannot grow in either direction.

Original entry on oeis.org

4, 0, 26, 0, 194, 17, 1086, 152, 6140, 1006, 32713, 6335, 172379, 38800

Offset: 15

John Mason, Jan 05 2023

			a(15) = 4 because of:
OOO   OOO   OOO   OOO
O O   O O   O O   O O
O  OO O  OO O  OO OO OO
O   O OO  O OOO O  OO O
OOOOO  OOOO   OOO   OOO

Cf. A002013.

cp's OEIS Frontend

A131482 a(n) is the number of n-celled polyominoes with perimeter 2n+2.

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A151514 Number of 1-sided snake polyominoes with n cells.

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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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A333313 Number of 2-sided strip polyominoes with n cells and no holes.

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A359706 Number of free (2-sided) ouroboros polyominoes with k=2n cells.

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A359521 Number of free mature snake polyominoes of n cells, where mature means that the snake cannot grow in either direction.

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