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A194276
Number of distinct polygonal shapes after n-th stage in the D-toothpick structure of A194270.
6
0, 0, 0, 0, 1, 3, 4, 5, 6, 7, 9, 10, 10, 11, 13, 13, 14
OFFSET
0,6
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?
EXAMPLE
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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Omar E. Pol, Aug 23 2011
STATUS
approved