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%I #29 Feb 24 2021 02:48:18
%S 0,1,3,7,13,21,31,43,57,69,81,99,123,153,183,211,241,261,273,291,317,
%T 351,393,443,499,553,597,645,709,791,871,939,1005,1041,1053,1071,1097,
%U 1131,1173,1223,1281,1339,1393,1459,1549,1663,1789,1911,2031,2133,2193
%N V-toothpick (or honeycomb) sequence (see Comments lines for definition).
%C A V-toothpick is constructed from two toothpicks of length 1 with a 120-degree angle between them, forming a V.
%C On the infinite hexagonal grid, we start at round 0 with no V-toothpicks.
%C At round 1 we place a V-toothpick anywhere in the plane.
%C 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.
%C At round 3 we place four other V-toothpicks.
%C And so on...
%C The structure looks like an unfinished honeycomb.
%C The sequence gives the number of V-toothpicks after n rounds. A161207 (the first differences) gives the number added at the n-th round.
%C See the entry A139250 for more information about the growth of toothpicks.
%C 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).
%C In the structure we can see distinct closed polygonal regions with side length equal to 1, for example: regular hexagons, concave decagons, concave dodecagons.
%H David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp120.jpg">Illustration of initial terms of A160120, A161206, A161328, A161330.</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%Y Cf. A139250, A160120, A160172, A161207, A161328, A161330.
%K nonn
%O 0,3
%A _Omar E. Pol_, Jun 08 2009
%E Terms beyond a(19) from _R. J. Mathar_, Jan 21 2010
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