

A182632


Toothpick sequence on the hexagonal net starting from a node.


15



0, 3, 9, 21, 33, 45, 69, 105, 129, 141, 165, 213, 273, 321, 381, 465, 513, 525, 549, 597, 657, 717, 801, 933, 1065, 1137, 1197, 1317, 1485, 1629, 1785, 1977, 2073, 2085, 2109, 2157, 2217, 2277, 2361, 2493, 2625, 2709, 2793, 2949, 3177, 3405, 3633
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OFFSET

0,2


COMMENTS

A connected network of toothpicks is constructed by the following iterative procedure. At stage 1, place three toothpicks each of length 1 on a hexagonal net, as a propeller, joined at a node. At each subsequent stage, add two toothpicks (which could be called a single Vtoothpick with a 120degree corner) adjacent to each node which is the endpoint of a single toothpick.
The exposed endpoints of the toothpicks of the old generation are touched by the endpoints of the toothpicks of the new generation. In the graph, the edges of the hexagons become edges of the graph, and the graph grows such that the nodes that were 1connected in the old generation are 3connected in the new generation.
It turns out heuristically that this growth does not show frustration, i.e., a free edge is never claimed by two adjacent exposed endpoints at the same stage; the rule of growing the network does apparently not need specifications to address such cases.
The sequence gives the number of toothpicks in the toothpick structure after nth stage. A182633 (the first differences) gives the number of toothpicks added at nth stage.
a(n) is also the number of components after nth stage in a toothpick structure starting with a single Ytoothpick in stage 1 and adding only Vtoothpicks in stages >= 2. For example: consider that in A161644 a Vtoothpick is also a polytoothpick with two components or toothpicks and a Ytoothpick is also a polytoothpick with three components or toothpicks. For more information about this comment see A161206, A160120 and A161644.
Assume that every triangular cell has area 1.
It appears that the structure contains only three types of polygons:
 Regular hexagons of area 6.
 Concave decagons (or concave 10gons) of area 12.
 Concave dodecagons (or concave 12gons) of area 18.
There are infinitely many of these polygons.
The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.
The structure has internal growth.
For an animation see the movie version in the Links section.
The animation shows the fractallike behavior the same as in other members of the family of toothpick cellular automata.
For another version starting with a simple toothpick see A182840.
For a version of the structure in the first quadrant but on the square grid see A182838. (End)


LINKS



FORMULA



EXAMPLE

a(0)=0. At stage 1 we place 3 toothpicks connected to the initial grid point of the structure. Note that there are 3 exposed endpoints. At stage 2 we place 6 toothpicks, so a(2)=3+6=9, etc.


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



