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

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

Offset

1,3

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.

Links

Table of n, a(n) for n=1..8.

Pontus von Brömssen, Illustration for sizes 1..4.

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).

Sequence in context: A286077 A286072 A002830 * A370096 A196015 A332625

Adjacent sequences: A365651 A365652 A365653 * A365655 A365656 A365657

Keyword

nonn,hard,more

Author

Pontus von Brömssen, Sep 17 2023

Status

approved