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

Author

Peter Kagey, Apr 20 2021

Keywords

• nonn
• more

Comments

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.

Links

• 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

Crossrefs

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).

Extensions

a(11) from Drake Thomas, May 02 2021

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