

A161206


Vtoothpick (or honeycomb) sequence (see Comments lines for definition).


26



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

0,3


COMMENTS

A Vtoothpick is constructed from two toothpicks of length 1 with a 120degree angle between them, forming a V.
On the infinite hexagonal grid, we start at round 0 with no Vtoothpicks.
At round 1 we place a Vtoothpick anywhere in the plane.
At round 2 we place two other Vtoothpicks. Note that, after round 2, in the structure there are three Vtoothpicks, with seven 120degree angles and one 240degree angle.
At round 3 we place four other Vtoothpicks.
And so on...
The structure looks like an unfinished honeycomb.
The sequence gives the number of Vtoothpicks after n rounds. A161207 (the first differences) gives the number added at the nth round.
See the entry A139250 for more information about the growth of toothpicks.
Note that, on the infinite hexagonal grid, a Vtoothpick can be represented as a polyedge with two components. In this case, at nth 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.


LINKS

Table of n, a(n) for n=0..50.
David Applegate, The movie version
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata


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


KEYWORD

nonn


AUTHOR

Omar E. Pol, Jun 08 2009


EXTENSIONS

Terms beyond a(19) from R. J. Mathar, Jan 21 2010


STATUS

approved



