A365654 Number of free n-polyominoids, allowing right-angled connections only ("hard" polyominoids).

1, 1, 5, 16, 90, 537, 3826, 28655, 225534

Offset: 1

Author

Pontus von Brömssen, Sep 17 2023

Keywords

• nonn
• hard
• more

Comments

Two squares are allowed to meet in a straight 180-degree connection, but the structure must be connected through right-angled ("hard") connections only. This seems to be in agreement with the definition of "hard" polyominoids in the Mireles Jasso link (the number of fixed hard hexominoids given by the "sample report" linked from that web-page agrees with A365655(6) = 22417), but differs from the definition in the Wikipedia article. The smallest example of a polyominoid that is included here but is not hard according to Wikipedia consists of two squares between (0,0,1) and (2,1,1), two between (0,0,1) and (2,0,2), and one between (1,0,0) and (1,1,1) (a "one-legged sofa", see illustration in the Mireles Jasso link). This explains why a(5) = 90, while the number of hard pentominoids is 89 according to the Wikipedia article.

Equivalently, number of n-polysticks in 3 dimensions, connected through right-angled connections.

Also, the number of face-connected polyhedral components in the square bipyramidal honeycomb up to translation, rotation, and reflection of the honeycomb. - Peter Kagey, Jun 10 2025

Links

• Pontus von Brömssen, Illustration for sizes 1..4.
• Peter Kagey, Animated illustration of the a(4)=16 free polyforms on the square bipyramidal honeycomb.
• Jorge Luis Mireles Jasso, The Polyominoids.
• Wikipedia, Polyominoid.
• Wikipedia, Polystick.
• Index entries for sequences related to polyominoes.

Crossrefs

13th and 17th row of A366766.

Cf. A075679 (polyominoids), A365559 (polysticks in 3 dimensions), A365655 (fixed).

Extensions

a(9) from Pontus von Brömssen, Mar 03 2025