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A161206
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V-toothpick (or honeycomb) sequence (see Comments lines for definition).
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23
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| A V-toothpick is constructed from two half-toothpicks with a 120 degree angle between them, forming a V.
On the infinite triangular 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 a 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 triangular 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.
In the structure we can see distinct closed polygonal regions with side length equal to 1, for example:
- Regular hexagons ..... (with 1 non covered grid point).
- Concave decagons ..... (with 2 non covered grid points).
- Concave dodecagons ... (with 4 non covered grid points).
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LINKS
| David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata
O. E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks) [From Omar E. Pol, Dec 06 2009]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences [From Omar E. Pol (info(AT)polprimos.com), Dec 06 2009]
Index entries for sequences related to cellular automata [From Omar E. Pol, Dec 06 2009]
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CROSSREFS
| Cf. A139250, A160120, A160172, A161207, A161328, A161330.
Sequence in context: A169627 A102508 A115298 * A025728 A084537 A002061
Adjacent sequences: A161203 A161204 A161205 * A161207 A161208 A161209
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KEYWORD
| nonn
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AUTHOR
| Omar E. Pol (info(AT)polprimos.com), Jun 08 2009
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EXTENSIONS
| Terms beyond a(19) from R. J. Mathar, Jan 21 2010
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