A121197 -id:A121197 - cp's OEIS frontend

A343577 Number of generalized polyforms on the truncated square tiling with n cells.

1, 2, 2, 7, 22, 93, 413, 2073, 10741, 57540, 312805, 1722483, 9564565, 53489304, 300840332, 1700347858, 9650975401

Offset: 0

Peter Kagey, Apr 20 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.

a(n) >= A343417(n), the number of (n-k)-polyominoes with k distinguished vertices.

Kadon Enterprises, Interesting solutions: Ochominoes.
Peter Kagey, Haskell program for computing sequence.
Peter Kagey, The a(3) = 7 generalized polyforms on the truncated square tiling with 3 cells.
Peter Kagey, The a(5) = 93 generalized polyforms on the truncated square tiling with 5 cells.
John Mason, Illustration of equivalence between truncated square polyforms and a mixture of crosses and squares.
Wikipedia, Truncated Square Tiling

Cf. A121197 (one-sided).

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).

a(11) from Drake Thomas, May 02 2021

a(12)-a(16) from John Mason, Mar 20 2022

A121195 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 n big squares and n small squares.

Original entry on oeis.org

1, 12, 152, 2538, 45084, 851717

Offset: 1

N. J. A. Sloane, Aug 17 2006

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

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

Livio Zucca, PolyMultiForms
Livio Zucca, The 12 2-BiSquares (from the Zucca web site).

Cf. A121196, A121197, A121198.

More terms from Don Reble, Aug 17 2007

A121196 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 n tiles, each composed of a big square and an adjacent little square.

Original entry on oeis.org

1, 11, 114, 1519, 20769

Offset: 1

N. J. A. Sloane, Aug 17 2006

nonn

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

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

Livio Zucca, PolyMultiForms

Cf. A121195, A121197, A121198.

Better definition from Don Reble, Aug 17 2007

cp's OEIS Frontend

A343577 Number of generalized polyforms on the truncated square tiling with n cells.

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A121195 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 n big squares and n small squares.

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Extensions

A121196 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 n tiles, each composed of a big square and an adjacent little square.

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Extensions