A006074 -id:A006074 - cp's OEIS frontend

A057433 Erroneous version of A006074.

1, 3, 4, 14, 30, 107, 318, 1106, 3671

Offset: 1

dead

M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 151.

A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

1, 2, 2, 6, 8, 22, 42, 112, 252, 650, 1584, 4091, 10369, 26938, 69651, 182116, 476272, 1253067, 3302187, 8733551, 23142116, 61477564, 163612714, 436278921, 1165218495, 3117021788, 8349892686, 22397754046, 60153261611

Offset: 1

Joseph Myers, Oct 15 2011

See the link below for a definition of the tetrakis square tiling. When a square grid cell is divided into triangles, it must be divided dexter (\) or sinister (/) according to the parity of the grid cell.

			For n=3 there are 4 triaboloes.  Of these, 2 conform to the tetrakis grid.  Each of these 2 has a unique dissection into 6 tetrakis cells. - _George Sicherman_, Mar 25 2021

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.

Peter Kagey, Example illustrating a(4) = 6.
Wikipedia, Tetrakis square tiling

Cf. A197466, A197467, A006074.

Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).

Name clarified by George Sicherman, Mar 25 2021
a(21)-a(26) from Aaron N. Siegel, May 18 2022
a(27)-a(29) from Bert Dobbelaere, Jun 04 2025

A245676 Number of convex polyaboloes (or convex polytans): number of distinct convex shapes that can be formed with n congruent isosceles right triangles. Reflections are not counted as different.

Original entry on oeis.org

1, 3, 2, 6, 3, 7, 5, 11, 5, 10, 7, 14, 7, 16, 11, 20, 9, 17, 13, 22, 12, 25, 18, 27, 14, 24, 20, 31, 18, 36, 26, 37, 19, 34, 28, 38, 24, 45, 34, 47, 26, 41, 36, 49, 35, 61, 44, 54, 32, 54, 45, 56, 40, 71, 56, 63, 40, 66, 56, 72, 49, 86, 66, 76, 51, 74, 67, 77

Offset: 1

Eli Fox-Epstein, Jul 29 2014

nonn

Side numbers range from 3 to 8. See Wang and Hsiung (1942). - Douglas J. Durian, Sep 24 2017

			For n=3, there are two trapezoids.

Douglas J. Durian, Table of n, a(n) for n = 1..750
Douglas J. Durian, Illustration of shapes for n=1..20.
Eli Fox-Epstein, Ryuhei Uehara, The Convex Configurations of "Sei Shonagon Chie no Ita" and Other Dissection Puzzles, arXiv:1407.1923 [cs.CG], (8-July-2014)
Eli Fox-Epstein, Kazuho Katsumata, Ryuhei Uehara, The Convex Configurations of “Sei Shonagon Chie no Ita,” Tangram, and Other Silhouette Puzzles with Seven Pieces, Institute of Electronics, Information Communication Engineers - Transactions on Fundamentals, E99-A (2016), 1084-1089.
Paul Scott, Convex Tangrams, Australian Mathematics Teacher, 62 (2006), 2-5. Confirms a(16)=20.
Fu Traing Wang and Chuan-Chih Hsiung, A Theorem on the Tangram, American Mathematical Monthly, 49 (1942), 596-599. Proves a(16)=20 and that convex polyabolos have no more than eight sides.
Douglas J. Durian, Description of shapes for n = 1..750

Strictly less than A006074 for n > 2.

a(n) = A093709(n) + A292146(n) + A292147(n) + A292148(n) + A292149(n) + A292150(n). [Wang and Hsiang (1942)] - Douglas J. Durian, Sep 24 2017

Definition clarified by Douglas J. Durian, Sep 24 2017

a(51) and beyond from Douglas J. Durian, Jan 24 2020

A354380 Number of free pseudo-polytans with n cells.

Original entry on oeis.org

1, 10, 91, 1432, 23547, 416177, 7544247, 139666895, 2623895224

Offset: 1

Aaron N. Siegel, May 24 2022

A pseudo-polytan is a planar figure consisting of n isosceles right triangles joined either edge-to-edge or corner-to-corner, in such a way that the short edges of the triangles coincide with edges of the square lattice. Two figures are considered equivalent if they differ only by a rotation or reflection.

The pseudo-polytans are constructed in the same way as ordinary polytans (A006074), but allowing for corner-connections. Thus they generalize polytans in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes (A000105).

			a(2) = 10, because there are 10 ways of adjoining two isosceles right triangles: 3 distinct edge-to-edge joins (cf. A006074), and 7 distinct corner-to-corner joins.

Aaron N. Siegel, Illustration showing a(2) = 10. The color of each figure corresponds to its number of symmetries.

Cf. A006074, A000105, A030222.

A151519 Number of 1-sided polytans (polyaboloes) with n cells.

Original entry on oeis.org

1, 4, 6, 22, 56, 198, 624, 2182, 7448, 26319, 92826, 332181, 1192845, 4315845, 15678200, 57227380, 209623109, 770516966, 2840466846, 10499678185, 38905008340, 144475534207

Offset: 1

Ed Pegg Jr, May 13 2009

Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polytan

Cf. A006074.

a(13)-a(14) from George Sicherman, Sep 27 2012
a(15) from George Sicherman, Aug 06 2013
a(15) corrected and a(16)-a(20) from John Mason, Jan 07 2022
a(21)-a(22) from Aaron N. Siegel, Jun 07 2022

A354382 Number of free pseudo-polyarcs with n cells.

Original entry on oeis.org

2, 32, 700, 21943, 737164, 25959013, 938559884

Offset: 1

Aaron N. Siegel, May 24 2022

See A057787 for a description of polyarcs. The pseudo-polyarcs are constructed in the same way as ordinary polyarcs, but allowing for corner-connections. Thus they generalize polyarcs in the same way that pseudo-polyominoes (aka polyplets, A030222) generalize ordinary polyominoes (A000105). They can also be viewed as the "rounded" variant of pseudo-polytans (A354380), in the same way that ordinary polyarcs are the rounded variant of ordinary polytans (A006074).
Two figures are considered equivalent if they differ only by a rotation or reflection.
The pseudo-polyarcs grow tremendously fast, much faster than most polyforms. The initial data that have been computed suggest an asymptotic growth rate of at least 36^n.

			a(10) = 32, because there are 32 ways of adjoining two monarcs: 7 distinct edge-to-edge joins, and 25 distinct corner-to-corner joins (including one double-corner join involving two concave arcs).

Aaron N. Siegel, Illustration showing a(2) = 32. The color of each figure corresponds to its number of symmetries.

Cf. A057787, A354380, A006074, A000105, A030222.

A337867 Number of polyfetts (or polifetti) with n cells, identifying mirror images.

Original entry on oeis.org

1, 10, 90, 1414, 23136, 406093, 7303813, 134027098

Offset: 1

George Sicherman, Sep 27 2020

A polyfett is a generalized polyabolo (or polytan). Its cells are equal isosceles right triangles on the quadrille grid, which may be joined along equal edges or at vertices.

Polyfetts are to polyaboloes what polyplets (or polykings) are to polyominoes.

			For n = 2, a(2) = 10. Two polyabolo cells can be joined at edges to form 3 different diaboloes, or at corners to form 7 different proper difetts.

Sean A. Irvine, Java program (github)
George Sicherman, Catalogue of Polyfetts
George Sicherman, polyaboloes.c

Cf. A006074.

/* See link to Unix C program polyaboloes.c under LINKS. */

A151520 Number of 1-sided strip polytans with n cells.

Original entry on oeis.org

1, 4, 6, 21, 47, 134, 323, 876, 2224, 5885, 15146, 39574, 102250, 265574, 686208, 1775646, 4583718, 11835037, 30515520, 78652647

Offset: 1

Ed Pegg Jr, May 13 2009

Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polytan

Cf. A006074, A151521.

a(6)-a(20) from John Mason, Jan 07 2022

A151521 Number of 2-sided strip polytans with n cells.

Original entry on oeis.org

1, 3, 4, 13, 25, 72, 166, 450, 1124, 2973, 7603, 19865, 51199, 132982, 343290, 888309, 2292325, 5918742, 15258937, 39329418

Offset: 1

Ed Pegg Jr, May 13 2009

Ed Pegg, Jr., Illustrations of polyforms
Eric Weisstein's World of Mathematics, Polytan

Cf. A006074, A151520.

a(6)-a(20) from John Mason, Jan 07 2022

A353978 Number of fixed polytans (polyaboloes) with n cells.

Original entry on oeis.org

4, 9, 24, 71, 224, 740, 2496, 8565, 29792, 104701, 371304, 1326702, 4771380, 17256161, 62712800, 228883359, 838492436, 3081972336, 11361867384, 41998361480, 155620033360, 577900838281

Offset: 1

Aaron N. Siegel, May 12 2022

Cf. A001168, A006074, A151519.

a(21)-a(22) from Aaron N. Siegel, Jun 07 2022

cp's OEIS Frontend

A057433 Erroneous version of A006074.

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A197465 Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

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A245676 Number of convex polyaboloes (or convex polytans): number of distinct convex shapes that can be formed with n congruent isosceles right triangles. Reflections are not counted as different.

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A354380 Number of free pseudo-polytans with n cells.

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A151519 Number of 1-sided polytans (polyaboloes) with n cells.

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A354382 Number of free pseudo-polyarcs with n cells.

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A337867 Number of polyfetts (or polifetti) with n cells, identifying mirror images.

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A151520 Number of 1-sided strip polytans with n cells.

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A151521 Number of 2-sided strip polytans with n cells.

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A353978 Number of fixed polytans (polyaboloes) with n cells.

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