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Number of diagonal polyominoes with n cells.
6

%I #64 Jun 01 2023 14:54:51

%S 1,1,2,5,15,54,212,908,4011,18260,84320,394462,1860872,8843896,

%T 42275308,203113670,980101070,4747504560,23074132601

%N Number of diagonal polyominoes with n cells.

%C Apparently the cells are circular blobs which must be connected diagonally and the polyominoes can be rotated by 90 degrees and turned over.

%C Also the number of essentially different (i.e., not related by reflections, translations or rotations) diagrams consisting of n nodes in Z^2 and n-1 horizontal or vertical edges of length 1 between pairs of nodes such that the resulting graph is connected (hence a tree). - _Paul Boddington_, Jul 27 2004

%C They are thus equivalent to a subset of the polyedges, counted by A019988, i.e., those that are treelike. - _John Mason_, Aug 20 2021

%C The number of treelike polyedges with n edges is a(n+1). - _John Mason_, Feb 12 2023

%H R. J. Mathar, <a href="/A056841/a056841.gz">Table of all such polyominoes with n <= 10 cells</a> (gzipped)

%H R. J. Mathar, <a href="/A056841/a056841.cpp.txt">C++ program</a>

%H Douglas A. Torrance, <a href="https://arxiv.org/abs/1906.01541">Enumeration of planar Tangles</a>, arXiv:1906.01541 [math.CO], 2019-2020. See Table 4.1 (C).

%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/d1.gif">The 15 5-celled diagonal polyominoes</a>

%H M. Vicher, <a href="/A056841/a056841.gif">The 15 5-celled diagonal polyominoes</a>

%F a(n+1) + A348095(n) = A019988(n). - _R. J. Mathar_, Sep 30 2021

%e The diagonal polyominoes with 1, 2, 3 and 4 cells are

%e O O O O O

%e \ \ \ /

%e O O O

%e \

%e O

%e O O O O O O

%e \ \ \ / \ / /

%e O O O O O O O

%e \ / \ \ / /

%e O O O O O

%e \ \

%e O O

%Y See also A056840, A056787, A019988 (free polysticks), A348095 (with cycles).

%K nonn,nice,more

%O 1,3

%A _James A. Sellers_, Aug 28 2000

%E Description revised by _N. J. A. Sloane_, Jun 21 2001

%E a(10) from _R. J. Mathar_, Apr 10 2006

%E a(11) from _Douglas A. Torrance_, Mar 06 2020

%E a(12)-a(14) from _John Mason_, Aug 14 2021

%E a(15)-a(19) from _John Mason_, Jun 01 2023