A151522 -id:A151522 - cp's OEIS frontend

A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

1, 2, 3, 9, 21, 68, 208, 730, 2542, 9287, 34053, 127112, 476849, 1803636, 6851960, 26157362, 100211446, 385239872, 1485232325, 5741327939, 22246061118, 86383655207, 336093789246, 1309999171971, 5114453234510, 19998176771431, 78306018629550, 307022197845116

Offset: 1

James Sellers, Aug 28 2000

John Mason, Table of n, a(n) for n = 1..50
R. J. Mathar, Illustration of shapes up to 7-ominoes
Ed Pegg, Jr., Illustrations of polyforms
M. Vicher, Polyforms
M. Vicher, The 21 5-celled rectangular polyominoes
M. Vicher, The 21 5-celled rectangular polyominoes
Eric Weisstein's World of Mathematics, Polyrect [From _Eric W. Weisstein_, Apr 24 2009]
Wikipedia, Polyomino

Cf. A000105 (cell is square), A151522 (1-sided), A001168 (fixed).

A[s_] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[Import[ "https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, } ][[All, 2]]];
A006749 = A@006749; A006746 = A@006746; A006748 = A@006748;
A006747 = A@006747; A056877 = A@056877; A056878 = A@056878;
A144553 = A@144553; A142886 = A@142886;
a[n_] := 2*A006749[[n]] + 2*A006746[[n]] + A006748[[n]] + 2*A006747[[n]] + 2*A056877[[n]] + A056878[[n]] + A144553[[n]] + A142886[[n + 1]];
Array[a, 28] (* Jean-François Alcover, Mar 26 2020 *)

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

Edited by N. J. A. Sloane, Apr 25 2001
Two more terms from Ed Pegg Jr, May 13 2009
a(13)-a(18) from Joseph Myers, Nov 15 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

cp's OEIS Frontend

A056780 Rectangular free polyominoes: number of n-celled polyominoes when the cell is a rectangle.

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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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