%I A161206
%S A161206 0,1,3,7,13,21,31,43,57,69,81,99,123,153,183,211,241,261,273
%N A161206 V-Toothpick (or honeycomb) sequence. (See Comments lines for definition).
%C A161206 A V-Toothpick is formed by two half toothpick with a 120 degree angle, 
               as a V.
%C A161206 On the infinite triangular grid, we start at round 0 with no V-Toothpicks.
%C A161206 At round 1 we place a V-Toothpick anywhere in the plane.
%C A161206 At round 2 we place two other V-Toothpicks. Note that, after round 2, 
               in the sieve there are three V-Toothpick, with seven 120 degree angles 
               and a 240 degree angle.
%C A161206 At round 3 we place four other V-Toothpicks.
%C A161206 And so on...
%C A161206 The sieve looks like an unfinished honeycomb.
%C A161206 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 A161206 See the entry A139250 for more information about the toothpick process 
               and the toothpick propagation.
%C A161206 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 
               sieve is a polyedge with 2*a(n) components.
%C A161206 In the sieve we can see distinct polygons with side length equal to 1, 
               for example:
%C A161206 - Regular hexagons ............ (with 1 non covered grid point).
%C A161206 - Concave-convex decagons ..... (with 2 non covered grid points).
%C A161206 - Concave-convex dodecagons ... (with 4 non covered grid points).
%Y A161206 Cf. A139250, A160120, A160172, A161207, A161328, A161330.
%Y A161206 Sequence in context: A025721 A102508 A115298 this_sequence A025728 A084537 
               A002061
%Y A161206 Adjacent sequences: A161203 A161204 A161205 this_sequence A161207 A161208 
               A161209
%K A161206 more,nonn
%O A161206 0,3
%A A161206 Omar E. Pol (info(AT)polprimos.com), Jun 08 2009