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A327330 "Concave pentagon" toothpick sequence (see Comments for precise definition).

%I A327330 #74 Dec 03 2019 07:15:35
%S A327330 0,1,3,7,11,15,23,33,41,45,53,63,75,89,111,133,149,153,161,171,183,
%T A327330 197,219,241,261,275,299,327,361,403,463,511,547,551,559,569,581,595,
%U A327330 617,639,659,673,697,725,759,801,861,909,949,967,995,1029,1075,1125,1183,1233,1281,1321,1389,1465,1549,1657
%N A327330 "Concave pentagon" toothpick sequence (see Comments for precise definition).
%C A327330 This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks (A160164) and V-toothpicks (A161206).
%C A327330 The surprising fact is that after 2^k stages the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two quadrilaterals (Q1 and Q2), both with their largest sides in vertical position, as shown below:
%C A327330 .
%C A327330                   *                 *
%C A327330                   *  *           *  *
%C A327330                   *     *     *     *
%C A327330                   *        *        *
%C A327330                   *   Q1   *   Q2   *
%C A327330                   *       * *       *
%C A327330                   *      *   *      *
%C A327330                   *     *     *     *
%C A327330                   *    *       *    *
%C A327330                   *   *    E    *   *
%C A327330                   *  *           *  *
%C A327330                   * *             * *
%C A327330                   **               **
%C A327330                   * * * * * * * * * *
%C A327330 .
%C A327330 Note that for n >> 1 both quadrilaterals look like right triangles.
%C A327330 Every polygon has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
%C A327330 For the construction of the sequence the rules are as follows:
%C A327330 On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
%C A327330 At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
%C A327330 For the next n generation we have that:
%C A327330 If n is even then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.
%C A327330 If n is odd then we add I-toothpicks in vertical position (see the example).
%C A327330 a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
%C A327330 A327331 (the first differences) gives the number of elements added at the n-th stage.
%C A327330 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
%C A327330 The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.
%C A327330 The structure is almost identical to the structure of A327332, but a little larger at the upper edge.
%C A327330 The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.
%C A327330 The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
%C A327330 For another version, very similar, starting with a V-toothpick, see A327332, which it appears that shares infinitely many terms with this sequence.
%H A327330 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 A327330 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A327330 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A327330 Conjecture: a(2^k) = A327332(2^k), k >= 0.
%e A327330 Illustration of initial terms:
%e A327330 .
%e A327330                |      /|\     |/|\|
%e A327330                |       |      | | |
%e A327330                       / \     |/ \|
%e A327330                               |   |
%e A327330 n   :  0       1       2        3
%e A327330 a(n):  0       1       3        7
%e A327330 After three generations there are five I-toothpicks and two V-toothpicks in the structure, so a(3) = 5 + 2 = 7 (note that in total there are 2*a(3) = 2*7 = 14 single toothpicks of length 1).
%Y A327330 First differs from A231348 at a(11).
%Y A327330 Cf. A047999, A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327331 (first differences), A327332 (another version).
%Y A327330 For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.
%K A327330 nonn
%O A327330 0,3
%A A327330 _Omar E. Pol_, Sep 01 2019