cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-29 of 29 results.

A349329 Number of polyominoes with 2n cells and exactly one line of reflection symmetry, where that one line is parallel to the grid and passes through the corner of at least one square.

Original entry on oeis.org

0, 0, 2, 7, 28, 100, 368, 1335, 4912, 18125, 67477, 252573, 951363, 3601113, 13695150, 52291510, 200389661, 770391542, 2970337861, 11482318605, 44491635790, 172766013959, 672185703574, 2619993338628
Offset: 1

Views

Author

John Mason, Nov 15 2021

Keywords

Examples

			a(3)=2 because of hexominoes:
  OO   and   O
  O          OO
  O          OO
  OO         O

Links

Crossrefs

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A349328.

Formula

a(n) = A006746(2n) - A349328(2n).
a(n) = 2*(A006747(n) + A006748(n)) + A144553(n) + A056878(n) + A006746(n) + 4*A006749(n). - corrected by John Mason, Feb 26 2023

Extensions

Name corrected by John Mason, Feb 01 2022

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

Views

Author

James Sellers, Aug 28 2000

Keywords

Comments

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

Links

Crossrefs

Cf. A000105, A056780, A057973, A151522.

Formula

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

Extensions

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

Views

Author

Ed Pegg Jr, May 13 2009

Keywords

Comments

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.

References

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

Links

Crossrefs

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)

Formula

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

Extensions

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

Views

Author

Joseph Myers, Nov 24 2010

Keywords

Comments

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.

References

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

Links

Crossrefs

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)

Formula

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

Extensions

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

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

Views

Author

Robert A. Russell, Nov 01 2021

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Views

Author

Robert A. Russell, Nov 01 2021

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A000988, A144553, A348848 (vertex center).
Inner rings: A324406, A324407, A324408, A324409.

A351191 Number of polyominoes of 4n cells with both horizontal and vertical symmetries, for which the 180-degree rotational symmetry has an axis that coincides with a vertex of a square of the underlying matrix, but without 90-degree rotational symmetry.

Original entry on oeis.org

0, 1, 2, 9, 29, 106, 372, 1356, 4928, 18200, 67541, 252847, 951615, 3602131, 13696141, 52295335, 200393559, 770406039, 2970353199, 11482373919, 44491696182, 172766226170, 672185941552, 2619994156474, 10228900153083, 39996337310835, 156612012801698
Offset: 1

Views

Author

John Mason, Feb 05 2022

Keywords

Comments

This sequence enumerates a subset of the polyominoes enumerated by A056877.

Examples

			a(2)=1 because of the 2 by 4 octomino.

Links

Crossrefs

Cf. A000105, A056877, A144553, A006747, A006746, A006749, A006748.

Formula

a(n)=A144553(n) + A056877(n) + 2 * A006747(n) + 2 * A006746(n) + 4 * A006749(n) + A006748(n).

A351142 Free chiral polyominoes with n squares, having 90-degree rotational symmetry about the center of a square of the underlying square matrix.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 3, 7, 0, 0, 14, 25, 0, 0, 55, 90, 0, 0, 216, 319, 0, 0, 816, 1136, 0, 0, 3098, 4088, 0, 0, 11711, 14868, 0, 0, 44462, 54526, 0, 0, 169213, 201527, 0, 0, 646441, 750169, 0, 0, 2477819, 2809931, 0, 0, 9530788, 10582370, 0, 0, 36768799, 40043785
Offset: 1

Views

Author

John Mason, Feb 02 2022

Keywords

Comments

The sequence enumerates a subset of the polyominoes enumerated by A144553.

Examples

			a(12)=1 because of:
   O
   OOOO
   O O
  OOOO
     O

Crossrefs

Cf. A000105, A144553, A234007.

Formula

For n multiple of 4, a(n) = A144553(n) - A234007(n/4); otherwise a(n) = A144553(n).

A357647 a(n) is the number of free unholey polyominoes of n cells with 90-degree rotational symmetry and no other.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 7, 7, 0, 0, 22, 24, 0, 0, 71, 82, 0, 0, 239, 280, 0, 0, 817, 970, 0, 0, 2841, 3403, 0, 0, 10027, 12064, 0, 0, 35800, 43193, 0, 0, 129007, 156011, 0, 0, 468541, 567664, 0, 0, 1713174
Offset: 1

Views

Author

John Mason, Oct 10 2022

Keywords

Crossrefs

Cf. A000104, A144553.
Previous Showing 21-29 of 29 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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