%N Toothpick sequence starting at the vertex of the outside corner of an infinite 120-degree wedge on hexagonal net.
%C Corner sequence for the toothpick structure on hexagonal net.
%C The sequence gives the number of toothpicks after n stages. A182837 (the first differences) gives the number added at the n-th stage. For more information see A182632 and A153006.
%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 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>
%e We start at stage 0 with no toothpicks.
%e At stage 1 we place a single toothpick touching a vertex of the infinite hexagon, in direction to the center of the hexagon, but on the outside corner, so a(1)=1.
%e At stage 2 we place 2 toothpicks touching the exposed endpoint of the initial toothpick, so a(2)=1+2=3.
%e At stage 3 we place 4 toothpicks, so a(3)=3+4=7.
%e At stage 4 we place 8 toothpicks, so a(4)=7+8=15.
%e At stage 5 we place 12 toothpicks, so a(5)=15+12=27.
%e After 5 stages the toothpick structure has 5 hexagons and 6 exposed endpoints.
%Y Cf. A139250, A153006, A182632, A182634, A182837, A182840.
%A _Omar E. Pol_, Dec 12 2010