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%I #34 Apr 12 2023 10:44:09
%S 1,2,5,22,99,580,3557,23295,155437,1057516,7271980,50478035,352901040
%N Number of rounded n-celled polyominoes.
%C There are n cells, drawn on a square grid, pointwise connected; polyominoes may be rotated by 90 degrees and turned over.
%C Comments from _Joseph Myers_, Oct 27 2003. "There is a figure for n=5 (the first term this differs from A030222) on the last Vicher's link. I think the following explains this sequence, but someone should do the computations to verify it (and probably compute counts for "fixed" shapes - orientation matters - and one-sided shapes - at the same time and add those sequences if not present).
%C "Consider a polyplet (A030222) as made up of n components which are polyominoes, those polyominoes being joined to each other only at corners. Then sever all but n-1 of the diagonal links in such a way that a spanning tree remains. The present sequence counts such spanning trees (where different orientations of the same spanning tree do not count as distinct; note that a single symmetrical polyplet can produce multiple identical spanning trees of lesser symmetry in different orientations, which count as the same).
%C "Similarly, A056841 appears to count spanning trees of polyominoes (ordinary polyominoes, A000105), where the edges shared by two squares are the edges of the graph for the purposes of forming the spanning tree and A056787 _may_ count spanning trees of polyplets where the graph has edges joining every pair of squares that share an edge or vertex (this definitely needs computations, but it does match the first three terms)."
%C The difference between this sequence and A030222 is illustrated through a comment and an image in A030222, also linked to here: the figures filled with identical color count as different here, but they represent the same polyplet and are counted only once in A030222. They all arise from adding one more square in three inequivalent positions (touching a corner, one side or two sides) to the (only) 4-polyplet with a hole (depicted here as not having a hole but rather a "bay", delimited to all but one (diagonal) direction). - _M. F. Hasler_, Sep 29 2014
%H Abaroth's World, <a href="https://abarothsworld.com/Puzzles/Polyominoes/Rounded%20Polyominoes.htm">Rounded Polyominoes</a>
%H M. F. Hasler, <a href="/A030222/a030222.gif">Colored illustration of A056840(5)=99 and A056840(5)=94 (based on Vicher's r7.gif)</a>.
%H M. Vicher, <a href="http://www.vicher.cz/puzzle/polyforms.htm">Polyforms</a>
%H M. Vicher, <a href="http://www.vicher.cz/puzzle/polyform/minio/images/r1.gif">The 22 4-celled rounded polyominoes</a> (<a href="/A056840/a056840.gif">local copy</a>).
%H M. Vicher, <a href="http://www.vicher.cz/puzzle/polyform/minio/images/r7.gif">The 99 5-celled rounded polyominoes</a>
%Y Cf. A030222, A056841, A000105, A362223, A362224.
%K nice,nonn,more
%O 1,2
%A _James A. Sellers_, Aug 28 2000
%E Edited by _N. J. A. Sloane_, Jun 21 2001
%E a(7)-a(13) from _John Mason_, Apr 12 2023
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