A056780 -id:A056780 - cp's OEIS frontend

A151522 Number of 1-sided polyrhombs with n cells.

1, 2, 4, 13, 35, 120, 392, 1405, 4998, 18378, 67792, 253509, 952534, 3604624, 13699554, 52304807, 200406370, 770442286, 2970401696, 11482513428, 44491881033, 172766765654, 672186650116, 2619996250930, 10228902882021, 39996345469572, 156612023354364, 614044364443761

Offset: 1

Ed Pegg Jr, May 13 2009

Also counts 1-sided polyrects.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polyrhomb
Eric Weisstein's World of Mathematics, Polyrect
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

A[n_] := With[{n6 = StringPadLeft[ToString[n], 6, "0"]}, Cases[ Import[ "https://oeis.org/A" <> n6 <> "/b" <> n6 <> ".txt", "Table"], {, }][[All, 2]]];
A006749 = A@6749; A006746 = A@6746; A006748 = A@6748; A006747 = A@6747; A056877 = A@56877; A056878 = A@56878; A144553 = A@144553; A142886 = A@142886;
a[n_] := 4*A006749[[n]] + 2*A006746[[n]] + 2*A006748[[n]] + 4*A006747[[n]] + 2*A056877[[n]] + 2*A056878[[n]] + 2*A144553[[n]] + A142886[[n + 1]];
a /@ Range[28] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = 4*A006749(n) + 2*A006746(n) + 2*A006748(n) + 4*A006747(n) + 2*A056877(n) + 2*A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A056783 Number of diamond polyominoes with n cells.

Original entry on oeis.org

1, 1, 3, 7, 20, 62, 204, 709, 2526, 9212, 33989, 126838, 476597, 1802618, 6850969, 26153537, 100207548, 385225375, 1485216987, 5741272625, 22246000726, 86383442996, 336093551268, 1309998354125, 5114452295933, 19998173607505, 78306014924606, 307022185565345

Offset: 1

James Sellers, Aug 28 2000

Also the number of polybricks of size n made of Lego.

John Mason, Table of n, a(n) for n = 1..50
Ed Pegg, Jr., Illustrations of polyforms.
M. Keller, Counting Polyforms.
M. Vicher, Polyforms.
Eric Weisstein's World of Mathematics, Polyrhomb.

Cf. A000105, A056780, A057973, A151522.

a(n) = 2*A006749(n) + A006746(n) + 2*A006748(n) + 2*A006747(n) + A056877(n) + 2*A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

More terms from Don Reble, Nov 01 2001
a(15)-a(18) from Joseph Myers, Nov 15 2010
Offset corrected and a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A151525 Number of poly-IH64-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 2, 4, 12, 35, 116, 392, 1390, 4998, 18321, 67791, 253288, 952527, 3603761, 13699516, 52301427, 200406183, 770429000, 2970400815, 11482461055, 44491876993, 172766558719, 672186631950, 2619995431640, 10228902801505, 39996342220199, 156612023001490, 614044351536722

Offset: 1

Ed Pegg Jr, May 13 2009

nonn

Equivalently, 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 polyrects, but that is A151522.

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

John Mason, Table of n, a(n) for n = 1..50
Jean-François Alcover, Mathematica program
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

a(n) = 4*A006749(n) + 3*A006746(n) + 2*A006748(n) + 2*A006747(n) + 2*A056877(n) + A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

Edited and a(13)-a(18) by Joseph Myers, Nov 24 2010

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A182645 Number of poly-IH68-tiles (holes allowed) with n cells.

Original entry on oeis.org

1, 1, 4, 10, 34, 110, 388, 1369, 4982, 18246, 67727, 253014, 952275, 3602743, 13698525, 52297602, 200402285, 770414503, 2970385477, 11482405741, 44491816601, 172766346508, 672186393972, 2619994613794, 10228901862928, 39996339056273, 156612019296546, 614044339256951

Offset: 1

Joseph Myers, Nov 24 2010

nonn

Equivalently, 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.

John Mason, Table of n, a(n) for n = 1..50
Jean-François Alcover, Mathematica program
Wikipedia, Polyomino

Polyominoes by group of symmetries relating shapes considered the same: A000105 (all symmetries), A001168 (translations only), A000988 (rotations and translations), A056780 (horizontal and vertical reflections, rotations of order 2 and translations), A056783 (reflections in either diagonal, rotations of order 2 and translations), A151522 (rotations of order 2 and translations), A151525 (reflections in a horizontal line and translations), A182645 (reflections in a NE-SW diagonal line and translations)

a(n) = 4*A006749(n) + 2*A006746(n) + 3*A006748(n) + 2*A006747(n) + A056877(n) + 2*A056878(n) + A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

a(19)-a(28) from Andrew Howroyd, Dec 04 2018

A385383 Irregular triangle read by rows: T(n,k) is the number of polyominoes of size k, i.e., connected subsets of k square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns; 1 <= k <= n^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 5, 6, 6, 3, 1, 1, 1, 2, 3, 9, 17, 44, 81, 150, 163, 161, 88, 56, 16, 8, 1, 1, 1, 2, 3, 9, 21, 62, 168, 490, 1324, 3370, 7433, 13905, 20961, 24927, 23008, 16766, 9825, 4669, 1831, 576, 157, 32, 8, 1, 1

Offset: 1

Pontus von Brömssen, Jun 27 2025

			Triangle begins:
  1;
  1, 2, 1, 1;
  1, 2, 3, 5,  6,  6,  3,   1,   1;
  1, 2, 3, 9, 17, 44, 81, 150, 163, 161, 88, 56, 16, 8, 1, 1;
  ...

Pontus von Brömssen, Illustration for row n = 3.

Cf. A056780, A385382 (row sums), A385385 (interchange of rows and columns of the torus allowed), A385388 (edge subsets).

T(n,k) = A056780(k) if n >= k.

T(n,k) <= 2*A385385(n,k), with equality if and only if k = 2.

A002369 Number of ways of folding a strip of n rectangular stamps.

Original entry on oeis.org

1, 2, 3, 8, 18, 44, 115, 294, 783

Offset: 1

N. J. A. Sloane

Is this an erroneous version of A056780? It is unclear why one of the 9 polyominoes counted in A056780(4) should be omitted here. Devisme writes in his article that errors are likely and he cannot guarantee the exact figures but only "the order of magnitude". - M. F. Hasler, Feb 24 2018

This is different from A056780 because the polyominoes can split off (beginning at A056780(4)) while a strip of stamps always has 2 ends. - Eric Fox, Sep 01 2019

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. Devisme, Contribution à l'étude du problème des timbres-poste, Sphinx, 7 (Dec. 1937), 202-203. [Annotated scanned copy. Keep this link because it shows the marginal notes.]
J. Devisme, Enhanced scan of first page
J. Devisme, Enhanced scan of second page
N. J. A. Sloane, Illustration of initial terms
Index entries for sequences obtained by enumerating foldings

Cf. A056780, A326301.

A326301 The number of rectangular free polyominoes with n cells where no 4 cells meet at a point.

Original entry on oeis.org

1, 2, 3, 8, 19, 54, 159, 511, 1676, 5685, 19512

Offset: 1

R. J. Mathar, Oct 17 2019

Obtained from the polyominoes of A056780 by discarding the polyominoes that contain 2 X 2 subblocks of 4 cells. One may call this the Tatami version of the free rectangular polyominoes.

This might also be considered a well-defined (or corrected?) version of A002369.

			From the A056780(4) = 9 polyominoes the one with the 2 X 2 bounding box is not counted, giving a(4)=8.
From the A056780(5)=21 polyominoes, 2 are not counted:
L
LL      and     LL
LL              LLL

Cf. A056780, A002369.

cp's OEIS Frontend

A151522 Number of 1-sided polyrhombs with n cells.

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A056783 Number of diamond polyominoes with n cells.

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A151525 Number of poly-IH64-tiles (holes allowed) with n cells.

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A182645 Number of poly-IH68-tiles (holes allowed) with n cells.

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A385383 Irregular triangle read by rows: T(n,k) is the number of polyominoes of size k, i.e., connected subsets of k square cells (or vertices), of the n X n flat torus, up to cyclic shifts and reflections of rows and columns; 1 <= k <= n^2.

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A002369 Number of ways of folding a strip of n rectangular stamps.

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A326301 The number of rectangular free polyominoes with n cells where no 4 cells meet at a point.

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