This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365654 #32 Jun 11 2025 01:01:05
%S A365654 1,1,5,16,90,537,3826,28655,225534
%N A365654 Number of free n-polyominoids, allowing right-angled connections only ("hard" polyominoids).
%C A365654 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.
%C A365654 Equivalently, number of n-polysticks in 3 dimensions, connected through right-angled connections.
%C A365654 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
%H A365654 Pontus von Brömssen, <a href="/A365654/a365654.svg">Illustration for sizes 1..4</a>.
%H A365654 Peter Kagey, <a href="/A365654/a365654.gif">Animated illustration of the a(4)=16 free polyforms on the square bipyramidal honeycomb</a>.
%H A365654 Jorge Luis Mireles Jasso, <a href="https://web.archive.org/web/20091026133059/http://www.geocities.com/jorgeluismireles/polyominoids/">The Polyominoids</a>.
%H A365654 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyominoid">Polyominoid</a>.
%H A365654 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polystick">Polystick</a>.
%H A365654 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%Y A365654 13th and 17th row of A366766.
%Y A365654 Cf. A075679 (polyominoids), A365559 (polysticks in 3 dimensions), A365655 (fixed).
%K A365654 nonn,hard,more
%O A365654 1,3
%A A365654 _Pontus von Brömssen_, Sep 17 2023
%E A365654 a(9) from _Pontus von Brömssen_, Mar 03 2025