%I A056840
%S A056840 1,2,5,22,99,580
%N A056840 Number of rounded n-celled polyominoes.
%C A056840 There are n cells, drawn on a square grid, pointwise connected; polyominoes 
               may be rotated by 90 degrees and turned over.
%C A056840 Comments from Joseph S. Myers (jsm(AT)polyomino.org.uk), Oct 27 2003. 
               "There is a figure for n=5 (the first term this differs from A030222) 
               on the web page http://alpha.ujep.cz/~vicher/puzzle/polyform/minio/
               images/r7.gif. 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 A056840 "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 A056840 "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)."
%H A056840 M. Vicher, <a href="http://www.vicher.cz/puzzle/polyforms.htm">Polyforms</
               a>
%H A056840 M. Vicher, <a href="http://www.vicher.cz/puzzle/polyform/minio/images/
               r1.gif">The 22 4-celled rounded polyominoes</a>
%H A056840 M. Vicher, <a href="a056840.gif">The 22 4-celled rounded polyominoes</
               a>
%H A056840 M. Vicher, <a href="http://www.vicher.cz/puzzle/polyform/minio/images/
               r7.gif">The 99 5-celled rounded polyominoes</a>
%Y A056840 Cf. A056841.
%Y A056840 Sequence in context: A041006 A083465 A030222 this_sequence A126797 A101206 
               A041807
%Y A056840 Adjacent sequences: A056837 A056838 A056839 this_sequence A056841 A056842 
               A056843
%K A056840 nice,nonn
%O A056840 1,2
%A A056840 James A. Sellers (sellersj(AT)math.psu.edu), Aug 28 2000
%E A056840 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jun 21 2001