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A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

%I A365995 #17 Mar 03 2025 17:45:24
%S A365995 1,2,9,66,691,9216,134325
%N A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.
%C A365995 This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.
%H A365995 Pontus von Brömssen, <a href="/A365995/a365995.svg">Illustration for sizes 1..3</a>.
%H A365995 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyominoid">Polyominoid</a>.
%H A365995 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%Y A365995 Cf. A365996 (fixed).
%Y A365995 21st row of A366766.
%Y A365995 The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.
%Y A365995   corner-connections | edge-connections |   free  |  fixed  | 1-sided
%Y A365995   -------------------+------------------+---------+---------+--------
%Y A365995          none        |         |        | A000105 |3*A001168| A000105
%Y A365995          none        |          L       | A365654 | A365655 |
%Y A365995          none        |         |L       | A075679 | A075678 | A056846
%Y A365995           |          |        none      | A000105 |3*A001168| A000105
%Y A365995           |          |         |        | A030222 |3*A006770| A030222
%Y A365995           |          |          L       | A365995 | A365996 |
%Y A365995           |          |         |L       | A365997 | A365998 |
%Y A365995            L         |        none      | A365999 | A366000 |
%Y A365995            L         |         |        | A366001 | A366002 |
%Y A365995            L         |          L       | A366003 | A366004 |
%Y A365995            L         |         |L       | A366005 | A366006 |
%Y A365995           |L         |        none      | A365652 | A365653 |
%Y A365995           |L         |         |        | A366007 | A366008 |
%Y A365995           |L         |          L       | A366009 | A366010 |
%Y A365995           |L         |         |L       | A365650 | A365651 |
%K A365995 nonn,hard,more
%O A365995 1,2
%A A365995 _Pontus von Brömssen_, Sep 26 2023
%E A365995 a(7) from _Pontus von Brömssen_, Mar 03 2025