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%I #54 Feb 22 2023 21:42:17
%S 0,1,5,13,27,43,57,81,119,151,165,189,235,299,353,409,495,559,573,597,
%T 643,707,769,849,975,1119,1205,1261,1371,1539,1697,1841,2039,2167,
%U 2181,2205,2251,2315,2377,2457,2583,2727,2821,2901,3043,3267,3505,3729,4015
%N Toothpick sequence on hexagonal net.
%C Rules:
%C - Each new toothpick must lie on the hexagonal net such that the toothpick endpoints coincide with two consecutive nodes.
%C - Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.
%C The sequence gives the number of toothpicks after n stages. A182841 (the first differences) gives the number added at the n-th stage.
%C The toothpick structure has polygons in which there are uncovered grid points, the same as A160120 and A161206. For more information see A139250.
%C Has a behavior similar to A151723, A182632. - _Omar E. Pol_, Feb 28 2013
%C From _Omar E. Pol_, Feb 17 2023: (Start)
%C Assume that every triangular cell has area 1.
%C It appears that the structure contains only three types of polygons:
%C - Regular hexagons of area 6.
%C - Concave decagons (or concave 10-gons) of area 12.
%C - Concave dodecagons (or concave 12-gons) of area 18.
%C There are infinitely many of these polygons.
%C The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.
%C For an animation see the movie version in the Links section.
%C The animation shows the fractal-like behavior the same as in other members of the family of toothpick cellular automata.
%C The structure has internal growth.
%C For another version starting from a node see A182632.
%C For a version of the structure in the first quadrant but on the square grid see A182838. (End)
%H Olaf Voß, <a href="/A182840/b182840.txt">Table of n, a(n) for n = 0..1000</a>
%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 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 Olaf Voß, <a href="http://oeis.org/wiki/Toothpick_structures_on_hexagonal_net">Illustration of initial terms</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>
%e We start at stage 0 with no toothpicks.
%e At stage 1 we place a toothpick anywhere in the plane (For example, in vertical position). There are two exposed endpoints, so a(1)=1.
%e At stage 2 we place 4 toothpicks. Two new toothpicks touching each exposed endpoint. So a(2)=1+4=5. There are 4 exposed endpoints.
%e At stage 3 we place 8 toothpicks. a(3)=5+8=13. The structure has 8 exposed endpoints.
%e At stage 4 we place 14 toothpicks (Not 16) because there are 4 endpoints that are touched by new 8 toothpicks but there are 4 endpoints that are touched by only 6 new toothpicks (not 8), so a(4)=13+14=27.
%e After 4 stages the toothpick structure has 4 hexagons and 8 exposed endpoints.
%Y Cf. A139250, A160120, A161206, A182632, A182634, A182838.
%K nonn
%O 0,3
%A _Omar E. Pol_, Dec 09 2010
%E More terms from _Olaf Voß_, Dec 24 2010
%E Wiki link added by _Olaf Voß_, Jan 14 2011
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