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%I #49 Sep 26 2015 01:17:13
%S 2,4,3,6,7,2,7,7,2,3,3,1,0,1,1,0,0,1,0,0,0,1
%N Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.
%C WARNING: The numbers are not fully tested. A new polygonal shape may appear in the structure beyond the stage 128 of A194270.
%C The cellular automaton of A194270 contains a large number of distinct polygonal shapes. For simplicity we call "polygons" to polygonal shapes.
%C In order to construct this sequence we use the following rules:
%C - Consider only the convex polygons and the concave polygons. Self-intersecting polygons are not counted (Note that some polygons contain in their body a toothpick or D-toothpick with an exposed endpoint; that element is not a part of the perimeter of the polygons).
%C - If two polygons have the same shape but they have different size then these polygons must be counted as distinct types of polygons.
%C - The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct types of polygons.
%C For more information see A194276 and A194278.
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e Consider toothpicks of length 2 and D-toothpicks of length sqrt(2):
%e a(3) = 2 because the structure contains 2 types of triangles, each with area: 1, 2.
%e a(4) = 4 because the structure contains 4 types of quadrilaterals: 3 squares, each with area: 2, 4, 8 and also a rectangle with area 8.
%e a(5) = 3 because the structure contains 3 types of pentagons: a concave pentagon with area = 3 and also 2 convex pentagons with area 5 and 6.
%e a(12) = 3 because the structure contains 3 types of dodecagons: a symmetric concave dodecagon with area 29 and also 2 asymmetrict concave dodecagons both with area = 18. These last dodecagons are essentially equal but with reflected shape, so a(12) = 3 not 2.
%Y Cf. A194270, A194276, A194278, A194444.
%K nonn,more,hard
%O 3,1
%A _Omar E. Pol_, Aug 25 2011
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