A000577 -id:A000577 - cp's OEIS frontend

A071333 Number of polyiamonds with n cells that do not tile the plane.

0, 0, 0, 0, 0, 0, 1, 0, 21, 107, 619, 1300, 6740, 19812, 61075, 181036, 577551, 1626183, 4904626, 14071461, 41336066, 120630534, 352200161, 1023454347, 2997105957, 8761810505, 25648786717, 75161609591, 220583137840, 647825855933

Offset: 1

Joseph Myers, May 19 2002

M. Gardner, Tiling with Polyominoes, Polyiamonds and Polyhexes. Chap. 14 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 175-187, 1988.

Joseph Myers, Polyiamond tiling

Equals A000577(n)-A071332(n) and A070764(n)+A071334(n). Cf. A054360, A070767.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000577 = A@000577;
A071332 = A@071332;
a[n_] := A000577[[n]] - A071332[[n]];
a /@ Range[30] (* Jean-François Alcover, Feb 21 2020 *)

More terms from Joseph Myers, Nov 11 2003

a(29) and a(30) from Joseph Myers, Nov 21 2010

A119602 Number of nonisomorphic polytetrahedra with n identical regular tetrahedra connected face-to-face or edge-to-edge (chiral shapes counted twice).

Original entry on oeis.org

1, 1, 2, 7, 39

Offset: 0

Jonathan Vos Post, Jun 02 2006

Polytetrahedra are a 3-dimensional generalization of polyiamonds, composed of unit regular tetrahedra in Euclidean 3-space. - Peter Kagey, Aug 05 2019, adapted from comment by Jonathan Vos Post.

			For n = 1, the a(1) = 1 polytetrahedron is the tetrahedron itself.
For n = 2, the a(2) = 2 polytetrahedra are formed by either gluing two tetrahedra along a face (triangular bipyramid) or gluing two tetrahedra along an edge.
For n = 7, the a(3) = 7 polytetrahedra are given in the links section.

Andrew I. Campbell, Valerie J. Anderson, Jeroen S. van Duijneveldt and Paul Bartlett, Dynamical Arrest in Attractive Colloids: The Effect of Long-Range Repulsion, Phys. Rev. Lett. 94, 208301 (2005).
Peter Kagey, Examples of the seven shapes that can be constructed from three tetrahedra, with Mathematica code.
J. F. Sadoc, Boerdijk-Coxeter helix and biological helices, Eur. Phys. J. B 12, 309-318.
Jonathan Vos Post, Original example for entry.
Eric Weisstein et al., Tetrahedron.
Wikipedia, Polyiamond
Wikipedia, Deltahedron

Cf. A000577, A000105.

A332930 Number of polyominoes with n cells on the order-6 square tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 2, 5, 16, 60, 255, 1193, 5862, 29875, 155995, 830539, 4489352, 24581675

Offset: 0

Peter Kagey, Mar 02 2020

The order-6 square tiling has Schläfli symbol {4,6}.

This sequence is computed from via program by Christian Sievers in the Code Golf Stack Exchange link.

Code Golf Stack Exchange, Impress Donald Knuth by counting polyominoes on the hyperbolic plane
Wikipedia, Order-6 square tiling
Wikipedia, Polyomino

Analogs with different Schläfli symbols are A000105 ({4,4}), A000207 ({3,oo}), A000577 ({3,6}), A005036 ({4,oo}), A119611 ({4,5}), and A330659 ({3,7}).

a(9)-a(14) from Ed Wynn, Feb 16 2021

A359689 a(n) is the number of free polyiamonds of n cells with chessboard coloring.

Original entry on oeis.org

2, 1, 2, 4, 8, 19, 48, 120, 320, 864, 2372, 6581, 18470, 52094, 147966, 421931, 1208214, 3470789, 10001186, 28892674, 83671476, 242823392, 706090582, 2056861981, 6001601254, 17538308071, 51323923796, 150389970179, 441211039118, 1295886198194, 3810209524640, 11214075937205, 33035791339150

Offset: 1

John Mason, Jan 11 2023

nonn

John Mason, Table of n, a(n) for n = 1..51
John Mason, Counting Polyiamonds
John Mason, Illustrations

Cf. A000577, A001933.

See link "Counting Polyiamonds".

More terms from John Mason, Oct 28 2023

A369365 Number of free rooted (or pointed) polyiamonds with n cells.

Original entry on oeis.org

1, 1, 2, 6, 16, 50, 150, 458, 1388, 4233, 12875, 39227, 119456, 363868, 1107912, 3372793, 10264156, 31228969

Offset: 1

Pontus von Brömssen, Jan 22 2024

Index entries for sequences related to polyominoes.

Cf. A000577, A126202, A369362, A369366.

a(18) from John Mason, Sep 20 2024

A369367 Number of free n-celled polyiamonds with the least number (A369366(n)) of inequivalent cells.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 27, 1, 1, 6, 2, 1, 523, 1

Offset: 1

Pontus von Brömssen, Jan 22 2024

Index entries for sequences related to polyominoes.

Cf. A000577, A367759, A369364, A369366.

a(18) from John Mason, Sep 20 2024

A385265 Number of edge-connected components of polygonal cells in the pinwheel tiling up to rotation of the tiling.

Original entry on oeis.org

1, 2, 4, 13, 53, 209, 904, 3963, 17900, 81745, 378554, 1768236, 8327789, 39471091, 188145066, 901117082, 4334151970, 20923370406, 101341800704, 492289834345

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "one-sided" polyforms because there are no reflectional symmetries of the pinwheel tiling.

Here the "pinwheel tiling" is a tiling consisting of rectangular and square cells, and does not refer to non-periodic triangular tilings.

Peter Kagey, Illustration of the pinwheel tiling.
Peter Kagey, Illustration of the a(4)=53 polyforms with 4 cells.

A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal), A344213 (truncated trihexagonal), A383908 (snub trihexagonal), A385266 (basketweave).

A385266 Number of edge-connected components of polygonal cells in the basketweave tiling up to rotation and reflection of the tiling.

Original entry on oeis.org

1, 2, 2, 10, 34, 166, 777, 4053, 21225, 114594, 624242, 3442399, 19121661, 106964679, 601639326, 3400619170, 19301719485, 109962791254

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

Each square cell in the basketweave tiling is edge-connected to four rectangular cells.

Peter Kagey, Illustration of the a(4)=34 polyforms with 4 cells.
Peter Kagey, Illustration of the basketweave tiling.

A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197459 (rhombille), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square), A344211 (rhombitrihexagonal), A344213 (truncated trihexagonal), A383908 (snub trihexagonal), A385265 (pinwheel).

A096361 Total number of triangles in all polyiamonds with n triangles.

Original entry on oeis.org

1, 2, 3, 12, 20, 72, 168, 528, 1440, 4480, 13046, 40008, 120055, 366324, 1109745, 3380752, 10269819, 31253904, 95011267, 288979680, 878550498, 2671223720, 8120041693, 24682865208, 75020015675, 227999634772, 692872971246, 2105464666676

Offset: 1

N. J. A. Sloane, Aug 04 2004

nonn

S. T. Coffin, The Puzzling World of Polyhedral Dissections, Chap 2, Table 1.

a(n) = n*A000577(n).

A130867 Triangular polyominoes (or polyiamonds) with n cells at most (turning over is allowed, holes are allowed, must be connected along edges).

Original entry on oeis.org

1, 2, 3, 6, 10, 22, 46, 112, 272, 720, 1906, 5240, 14475, 40641, 114624, 325921, 930028, 2666356, 7666949, 22115933, 63951671, 185370931, 538416222, 1566868939, 4567669566, 13336886288, 38998848186, 114194014853

Offset: 1

Tanya Khovanova, Jul 23 2007

nonn

Partial sums of A000577.

cp's OEIS Frontend

A071333 Number of polyiamonds with n cells that do not tile the plane.

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A119602 Number of nonisomorphic polytetrahedra with n identical regular tetrahedra connected face-to-face or edge-to-edge (chiral shapes counted twice).

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A332930 Number of polyominoes with n cells on the order-6 square tiling of the hyperbolic plane.

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A359689 a(n) is the number of free polyiamonds of n cells with chessboard coloring.

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A369365 Number of free rooted (or pointed) polyiamonds with n cells.

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A369367 Number of free n-celled polyiamonds with the least number (A369366(n)) of inequivalent cells.

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A385265 Number of edge-connected components of polygonal cells in the pinwheel tiling up to rotation of the tiling.

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A385266 Number of edge-connected components of polygonal cells in the basketweave tiling up to rotation and reflection of the tiling.

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A096361 Total number of triangles in all polyiamonds with n triangles.

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A130867 Triangular polyominoes (or polyiamonds) with n cells at most (turning over is allowed, holes are allowed, must be connected along edges).

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