A194277 Known number of distinct polygonal shapes with n sides in the infinite D-toothpick structure of A194270.

2, 4, 3, 6, 7, 2, 7, 7, 2, 3, 3, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1

Offset: 3

Author

Omar E. Pol, Aug 25 2011

Keywords

• nonn
• more
• hard

Comments

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.

For more information see A194276 and A194278.

Examples

			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.
		

Links

• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A194270, A194276, A194278, A194444.