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A161206
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V-toothpick (or honeycomb) sequence (see Comments lines for definition).
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34
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0, 1, 3, 7, 13, 21, 31, 43, 57, 69, 81, 99, 123, 153, 183, 211, 241, 261, 273, 291, 317, 351, 393, 443, 499, 553, 597, 645, 709, 791, 871, 939, 1005, 1041, 1053, 1071, 1097, 1131, 1173, 1223, 1281, 1339, 1393, 1459, 1549, 1663, 1789, 1911, 2031, 2133, 2193
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OFFSET
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0,3
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COMMENTS
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A V-toothpick is constructed from two toothpicks of length 1 with a 120-degree angle between them, forming a V.
On the infinite hexagonal grid, we start at round 0 with no V-toothpicks.
At round 1 we place a V-toothpick anywhere in the plane.
At round 2 we place two other V-toothpicks. Note that, after round 2, in the structure there are three V-toothpicks, with seven 120-degree angles and one 240-degree angle.
At round 3 we place four other V-toothpicks.
And so on...
The structure looks like an unfinished honeycomb.
The sequence gives the number of V-toothpicks after n rounds. A161207 (the first differences) gives the number added at the n-th round.
See the entry A139250 for more information about the growth of toothpicks.
Note that, on the infinite hexagonal grid, a V-toothpick can be represented as a polyedge with two components. In this case, at n-th round, the structure is a polyedge with 2*a(n) components (or 2*a(n) toothpicks).
In the structure we can see distinct closed polygonal regions with side length equal to 1, for example: regular hexagons, concave decagons, concave dodecagons.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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