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%I #48 May 16 2022 02:35:20
%S 1,1,3,4,12,27,78,208,635,1859,5726,17526,54620,170479,536714,1694567,
%T 5376764,17110286,54631302,174879997,561229678,1805022806
%N Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).
%C [3.6.3.6] refers to the face configuration of the rhombille tiling. - _Peter Kagey_, Mar 01 2020
%C If we draw the short diagonals of each tile in the rhombille tiling, we get a subset of edges of the regular hexagonal grid; two edges are adjacent if and only if the corresponding rhombi are adjacent. These are polyedges where angles are constrained to 120 degrees. So there is a 1-to-1 correspondence with the subset of polyedges counted in A159867 after removing polyedges with angles of 60 and/or 180 degrees. - _Joseph Myers_, Jul 12 2020
%C These are also known as polytwigs, by association with their representation as polyedges. - _Aaron N. Siegel_, May 15 2022
%D Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, Sections 2.7, 6.2 and 9.4.
%H Abaroth's World, <a href="http://abarothsworld.com/Puzzles/Polyforms/Polytwigs.htm">Illustration of a(1) through a(6)</a>
%H Peter Kagey, <a href="/A197459/a197459.pdf">Illustration for A197459(4) = 4</a>.
%H R. J. Mathar, <a href="/A197459/a197459_1.pdf">OEIS A197459</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rhombille_tiling">Rhombille tiling</a>
%Y Cf. A019988, A159867, A197460, A197461.
%Y Analogous for other tilings: A000105 (square), A000228 (hexagonal), A000577 (triangular), A197156 (prismatic pentagonal), A197159 (floret pentagonal), A197462 (kisrhombille), A197465 (tetrakis square), A309159 (snub square), A343398 (trihexagonal), A343406 (truncated hexagonal), A343577 (truncated square).
%K hard,nonn,more
%O 1,3
%A _Joseph Myers_, Oct 15 2011
%E a(18)-a(22) from _Aaron N. Siegel_, May 15 2022
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