A194276 Number of distinct polygonal shapes after n-th stage in the D-toothpick structure of A194270.

0, 0, 0, 0, 1, 3, 4, 5, 6, 7, 9, 10, 10, 11, 13, 13, 14

Offset: 0

Author

Omar E. Pol, Aug 23 2011

Keywords

• nonn
• hard
• more

Comments

The cellular automaton of A194270 contains a large number of distinct polygonal shapes. For simplicity we also call polygonal shapes "polygons".

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 polygon.)

- If two polygons have the same shape but they have different size then these polygons must be counted as distinct polygonal shapes.

- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct polygonal shapes.

For more information see A194277 and A194278.

Question: Is there a maximal record in this sequence?

Examples

			Consider toothpicks of length 2 and D-toothpicks of length sqrt(2).
.
Stage       New type  Perimeter    Area   Term       a(n)
. 0            -          -          -    a(0) =       0
. 1            -          -          -    a(1) =       0
. 2            -          -          -    a(2) =       0
. 3            -          -          -    a(3) =       0
. 4         hexagon   4*sqrt(2)+4    6    a(4) =       1
. 5   5.1   hexagon   2*sqrt(2)+8    8
.     5.2   octagon   4*sqrt(2)+8   14    a(5) = 1+2 = 3
. 6         pentagon  2*sqrt(2)+6    5    a(6) = 3+1 = 4
. 7         enneagon  6*sqrt(2)+6   13    a(7) = 4+1 = 5
		

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. A139250, A194270, A194271, A194277, A194278, A194444.