A343577 -id:A343577 - cp's OEIS frontend

A121197 Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from any n squares.

Original entry on oeis.org

2, 2, 8, 34, 158, 777, 4006, 21224, 114348, 624222, 3441050, 19121530, 106957272

Offset: 1

N. J. A. Sloane, Aug 17 2006

The Zucca web site calls these figures "n-DifferentSquares".

Also the number of one-sided polyforms on the faces of the truncated square tiling. - Peter Kagey, May 24 2025

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

Peter Kagey, Illustration of a(5)=158.
Livio Zucca, PolyMultiForms

Cf. A121195, A121196, A121198, A343577.

More terms from Don Reble, Aug 17 2007

a(13) from Joseph Myers, Oct 06 2011

A344213 Number of generalized polyforms on the truncated trihexagonal tiling with n cells.

Original entry on oeis.org

1, 3, 3, 14, 50, 261, 1397, 8364, 50643, 315512, 1984853, 12619579, 80802300, 520724842, 3373646060, 21959949924, 143527242317

Offset: 0

Peter Kagey, May 11 2021

This sequence counts "free" polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

Peter Kagey, Illustration of a(3) = 14.
Wikipedia, Truncated trihexagonal tiling

Analogous for other tilings: 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).

a(10)-a(16) from Bert Dobbelaere, Jun 07 2025

A383908 Number of generalized polyforms with n cells on the snub trihexagonal tiling.

Original entry on oeis.org

1, 3, 3, 7, 23, 69, 228, 766, 2642, 9309, 33382, 120629, 439752, 1613135, 5953061, 22075011, 82204128, 307213215, 1151820825, 4330858682, 16326297768, 61690058385

Offset: 0

Peter Kagey, May 14 2025

A generalized polyform on the snub trihexagonal tiling with n-cells is a collection of n faces connected edgewise. Two polyforms are considered the same they are related by an isometry (translation and/or rotation) of the snub trihexagonal tiling.

			For n=1, the a(1) = 3 generalized polyforms are the three types of faces: hexagons, hexagon-adjacent triangles, and hexagon-nonadjacent triangles.
For n=2, the a(2) = 3 generalized polyforms are
(1) a hexagon with a hexagon-adjacent triangle,
(2) a hexagon-adjacent triangle with a hexagon-nonadjacent triangle, and
(3) two hexagon-adjacent triangles.

Peter Kagey, Illustration of a(1)-a(4).
Wikipedia, Snub trihexagonal tiling

Analogous for other tilings: 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).

a(12)-a(21) from Bert Dobbelaere, Jun 05 2025

A343417 a(n) is the number of free polyominoes with k cells and n-k distinguished vertices.

Original entry on oeis.org

1, 1, 2, 6, 19, 71, 300, 1370, 6563, 32272, 161700, 820166, 4198764, 21647353, 112262033, 585049063, 3061951973, 16084816384, 84773694223

Offset: 0

Peter Kagey, Apr 15 2021

This sequence counts "free" polyominoes where holes are allowed. This means that two polyominoes are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.
A000105(n) <= a(n) <= A343577(n).
For an ordinary, asymmetrical polyomino, the number of free polyominoes with d distinguished cells is equal to C(v,d), where v is the number of vertices of the polyomino, and C is the binomial coefficient (A007318). - John Mason, Mar 11 2022

			For n = 3, the a(3) = 6 polyominoes with k cells and 3-k distinguished vertices are:
+---+                     *---+  +---+
|   |                     |   |  |   |
+   +---+  +---+---+---+  +   +  *   +  *---+  *---+
|       |  |           |  |   |  |   |  |   |  |   |
+---+---+, +---+---+---+, +---+, +---+, *---+, +---*,
where distinguished vertices are marked with asterisks.
For n = 4, a(4) = 19 because there are A000105(4) = 5 polyominoes with four cells and no distinguished vertices, 7 polyominoes with three cells and one distinguished vertex, 6 polyominoes with two cells and two distinguished vertices, and 1 polyomino with one cell and three distinguished vertices.

Peter Kagey, Haskell program for computing sequence.

Cf. A000105, A343577.

a(11)-a(18) from John Mason, Mar 11 2022

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).

A057707 a(n) is the number of free 'polyominoes' on the (4.8.8) grid with n octagons and n squares.

Original entry on oeis.org

1, 8, 80, 1296, 22601, 426305, 8340302, 167925635, 3453618607

Offset: 1

Warren Power (wjpnply(AT)hotmail.com), Oct 24 2000

This sequence enumerates a subset of the polyforms enumerated by A343577. - John Mason, Jan 15 2023

John Mason, Illustration for n = 2
Wikipedia, Truncated square tiling.
Livio Zucca, PolyMultiForms

Cf. A343577.

Better description from Larry Reeves (larryr(AT)acm.org), Nov 30 2000
a(6)-a(8) from John Mason, Jan 15 2023
a(9) from John Mason, Mar 01 2023

A345076 Number of generalized polyforms on the elongated triangular tiling with n cells.

Original entry on oeis.org

1, 2, 3, 5, 13, 32, 96, 283, 907, 2929, 9787, 32939, 112476, 386230, 1336150, 4642930, 16208851, 56786242, 199614651, 703678568, 2487109359, 8811020024, 31281360326, 111272475650

Offset: 0

Drake Thomas, Jun 07 2021

This sequence counts free polyforms where holes are allowed. This means that two polyforms are considered the same if one is a rigid transformation (translation, rotation, reflection or glide reflection) of the other.

			See the PDF in the links section.

Peter Kagey, The a(3) = 5 generalized polyforms on the elongated triangular tiling with 3 cells.
Wikipedia, Elongated Triangular Tiling

Analogous for other tilings: 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).

a(15)-a(23) from Bert Dobbelaere, Jun 05 2025

cp's OEIS Frontend

A121197 Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from any n squares.

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A344213 Number of generalized polyforms on the truncated trihexagonal tiling with n cells.

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A383908 Number of generalized polyforms with n cells on the snub trihexagonal tiling.

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A343417 a(n) is the number of free polyominoes with k cells and n-k distinguished vertices.

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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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A057707 a(n) is the number of free 'polyominoes' on the (4.8.8) grid with n octagons and n squares.

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A345076 Number of generalized polyforms on the elongated triangular tiling with n cells.

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