%I #68 Jun 18 2017 09:45:04

%S 1,1,3,18,139,1286,12715,130875

%N Number of inequivalent connected planar figures that can be formed from n 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree.

%C Figures that differ only by a rotation and/or reflection are regarded as equivalent (cf. A216581).

%C A216583 is A216492 without the condition that the adjacency graph of the dominoes forms a tree.

%C This is a subset of polydominoes. It appears that a(n) < A216583(n) < A056785(n) < A056786(n) < A210996(n) < A210988(n) < A210986(n), if n >= 3. - _Omar E. Pol_, Sep 15 2012

%H C. E. Lozada, <a href="/A216492/a216492.jpg">Illustration of initial terms: planar figures with up to 3 dominoes</a>

%H N. J. A. Sloane, <a href="/A056786/a056786.jpg">Illustration of initial terms of A056786, A216598, A216583, A216595, A216492, A216581</a> (Exclude figures marked (A) or (B))

%H N. J. A. Sloane, <a href="/A056786/a056786.pdf">Illustration of third term of A056786, A216598, A216583, A216595, A216492, A216581</a> (a better drawing for the third term)

%H M. Vicher, <a href="http://www.vicher.cz/puzzle/polyforms.htm">Polyforms</a>

%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>

%e One domino (2 X 1 rectangle) is placed on a table.

%e A 2nd domino is placed touching the first only in a single edge (length 1). The number of different planar figures is a(2)=3.

%e A 3rd domino is placed in any of the last figures, touching it and sharing just a single edge with it. The number of different planar figures is a(3)=18.

%e When n=4, we might place 4 dominoes in a ring, with a free square in the center. This is however not allowed, since the adjacency graph is a cycle, not a tree.

%Y Cf. A056786, A216598, A216583, A216595, A216492, A216581.

%Y Without the condition that the adjacency graph forms a tree we get A216583 and A216595.

%Y If we allow two long edges to meet we get A056786 and A216598.

%K nonn,more,nice

%O 0,3

%A _César Eliud Lozada_, Sep 07 2012

%E Edited by _N. J. A. Sloane_, Sep 09 2012