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A355310 Total number of V-toothpicks after n-th stage in a cellular automaton with V-toothpicks of 60 degrees (see Comments lines for precise definition).

%I A355310 #80 Aug 31 2022 09:12:31
%S A355310 0,1,3,7,13,21,27,37,51,69,79,89,103,123,141,165,201,245,267
%N A355310 Total number of V-toothpicks after n-th stage in a cellular automaton with V-toothpicks of 60 degrees (see Comments lines for precise definition).
%C A355310 An idea from Jean Hoffmann.
%C A355310 In this cellular automaton a V-toothpick is formed by 2 toothpicks of length 1 that share a vertex and the angle between both toothpicks is 60 degrees.
%C A355310 On the infinite triangular grid we start with no V-toothpick, so a(0) = 0.
%C A355310 At stage 1 we place a V-toothpick upside down, so a(1) = 1.
%C A355310 At every stage the V-toothpicks of the new generation must be connected to the structure by touching with their middle vertex the free ends of the V-toothpicks of the previous generation following a special rule:
%C A355310 The new V-toothpicks must be placed between the imaginary straight line containing the two extreme ends of the V-toothpick of the previous generation and the imaginary straight line that contains the middle vertex of that V-toothpick and that it is parallel to the aforementioned straight line.
%C A355310 A355311(n) gives the number of V-toothpicks added to the structure at the n-th stage.
%C A355310 2*a(n) is the total number of toothpicks of length 1 in the structure after n-th stage.
%C A355310 This cellular automaton is a companion of the Y-toothpick cellular automaton of A160120 in the sense that both essentially grow as an equilateral triangle.
%C A355310 This cellular automaton is slightly less symmetrical than Y-toothpick cellular automaton because its structure has a "backbone" formed by concave hexagons from the center of the triangle to one of its vertices.
%C A355310 The behavior could be very close to A160120 and similar to A153006 (see the graph).
%C A355310 After 18 stages we can see in the structure the following polygons:
%C A355310 - Equilateral triangles of perimeter 3.
%C A355310 - Equilateral triangles of perimeter 6 that contain 4 triangular cells.
%C A355310 - Concave hexagons of perimeter 8 that contain 6 triangular cells.
%C A355310 - Concave dodecagons (or concave 12-gons) of perimeter 18 that contain 22 triangular cells.
%H A355310 Jean Hoffmann and Omar E. Pol, <a href="/A355310/a355310_1.jpg">Illustration of a(18) = 267</a>
%H A355310 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 A355310 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A355310 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e A355310 Illustration of initial terms:
%e A355310 .
%e A355310                                                                   /__\
%e A355310                                                _\  /_            _\  /_
%e A355310                                  /__\           /__\            /\/__\/\
%e A355310             /\      _\/\/_      _\/\/_         _\/\/_          /__\/\/__\
%e A355310                                /\    /\     _\/\/__\/\/_      _\/\/__\/\/_
%e A355310                                                              /\          /\
%e A355310 .
%e A355310   n:         1         2           3              4                 5
%e A355310   a(n):      1         3           7             13                21
%e A355310 .
%Y A355310 Cf. A355311 (first differences).
%Y A355310 Cf. A139250, A153006, A160120, A161206, A161412, A161420, A299476, A299478, A327330, A327332.
%K A355310 nonn,more
%O A355310 0,3
%A A355310 Jean Hoffmann and _Omar E. Pol_, Jul 20 2022