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A194277
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Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.
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7
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2, 4, 3, 6, 7, 2, 7, 7, 2, 3, 3, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1
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OFFSET
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3,1
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COMMENTS
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WARNING: The numbers are not fully tested. A new polygonal shape may appear in the structure beyond the stage 128 of A194270.
The cellular automaton of A194270 contains a large number of distinct polygonal shapes. For simplicity we call "polygons" to polygonal shapes.
In order to construct this sequence we use the following rules:
- 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).
- If two polygons have the same shape but they have different size then these polygons must be counted as distinct types of polygons.
- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct types of polygons.
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LINKS
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EXAMPLE
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Consider toothpicks of length 2 and D-toothpicks of length sqrt(2):
a(3) = 2 because the structure contains 2 types of triangles, each with area: 1, 2.
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.
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.
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.
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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STATUS
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approved
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