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 21-26 of 26 results.

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

A056880 Number of n-celled polyominoes without holes, symmetric about axis 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 4, 6, 10, 11, 16, 18, 33, 34, 56, 56, 110, 106, 193, 177, 372, 335, 662, 571, 1269, 1084, 2282, 1875, 4365, 3568, 7914, 6257, 15141, 11935, 27615, 21170, 52911, 40486, 96980, 72490, 186144, 138981, 342638, 250814, 658884, 482029, 1217327, 875697, 2345272, 1686803
Offset: 1

Views

Author

N. J. A. Sloane, Sep 03 2000

Keywords

Comments

In other words, free polyominoes, without holes, with both vertical and horizontal symmetries but no other. - John Mason, Oct 10 2022

Links

Crossrefs

Cf. A000104, A056877 (with holes allowed).

Extensions

a(29)-a(52) from John Mason, Oct 10 2022

A351190 Number of polyominoes of n cells with both horizontal and vertical symmetries, for which the 180-degree rotational symmetry has an axis that coincides with the center of a square, but without 90-degree rotational symmetry.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 3, 0, 4, 1, 10, 3, 17, 6, 35, 15, 64, 31, 128, 74, 241, 145, 476, 332, 912, 649, 1789, 1446, 3456, 2821, 6779, 6164, 13161, 12049, 25839, 25981, 50348, 50844, 98957, 108514, 193375, 212766, 380522, 450568, 745248, 884843, 1468202, 1862065, 2880780, 3662652, 5681501, 7668922
Offset: 1

Views

Author

John Mason, Feb 04 2022

Keywords

Comments

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

Examples

			a(3)=1 because of the straight tromino.

Crossrefs

Cf. A000105, A056877.

Extensions

More terms from John Mason, Mar 15 2023

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).

A176673 Partial sums of A001168.

Original entry on oeis.org

1, 3, 9, 28, 91, 307, 1067, 3792, 13702, 50148, 185416, 691277, 2595167, 9800041, 37194707, 141787644, 542583488, 2083404030, 8024142706, 30988922366, 119972435149, 465505007827, 1809877343351, 7049866113619, 27507668129630
Offset: 1

Views

Author

Jonathan Vos Post, Apr 23 2010

Keywords

Comments

Partial sums of number of fixed polyominoes with n cells, for n > 0.

Examples

			a(6) = 1 + 2 + 6 + 19 + 63 + 216 = 307.

Links

Crossrefs

Cf. A001168, A000105, A006746, A056877, A006748, A056878, A006747, A006749, A006762.

Formula

a(n) = Sum_{i=1..n} A001168(i).

A343562 The number of n-cell polyominoes with a line of symmetry parallel to the edges.

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 12, 28, 44, 98, 157, 359, 583, 1324, 2183, 4961, 8263, 18729, 31477, 71231, 120605, 272380, 464188, 1046516, 1793505, 4036645, 6952368, 15623258, 27026985, 60645746, 105327495, 236021747, 411377265, 920654994, 1609862767, 3598552287
Offset: 1

Views

Author

R. J. Mathar, Sep 29 2021

Keywords

Comments

The polyominoes counted here may have additional symmetries (mirror axes along diagonals or rotational symmetries or two orthogonal mirror axes...). The mirror axes may run through cell edges or through cell centers.

Examples

			For n=4 the 3 polyominoes are (i) the 2x2 block, (ii) the straight 1x4 block and (iii) the T.

Links

Crossrefs

Cf. A056877, A142886, A006746.

Formula

a(n) = A056877(n) + A142886(n) + A006746(n).

Extensions

More terms from John Mason, Dec 16 2021
Previous Showing 21-26 of 26 results.

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

Content is available under The OEIS End-User License Agreement.