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A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).

%I A327332 #51 Dec 03 2019 07:16:36
%S A327332 0,1,3,7,11,15,21,33,41,45,51,63,75,85,101,133,149,153,159,171,183,
%T A327332 193,209,241,261,273,291,327,363,389,431,515,547,551,557,569,581,591,
%U A327332 607,639,659,671,689,725,761,787,829,913,953,969,993,1041,1085,1109,1149,1229,1277,1309,1357,1453,1549,1613
%N A327332 "Concave pentagon" toothpick sequence, starting with a V-toothpick (see Comments for precise definition).
%C A327332 Another version and very similar to A327330.
%C A327332 This arises from a hybrid cellular automaton on a triangular grid formed of V-toothpicks (A161206) and I-toothpicks (A160164).
%C A327332 After 2^k stages, the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two right triangles (R1 and R2) both with their hypotenuses in vertical position, as shown below:
%C A327332 .
%C A327332                   *                 *
%C A327332                   *  *           *  *
%C A327332                   *     *     *     *
%C A327332                   *        *        *
%C A327332                   *  R1   * *   R2  *
%C A327332                   *      *   *      *
%C A327332                   *     *     *     *
%C A327332                   *    *       *    *
%C A327332                   *   *    E    *   *
%C A327332                   *  *           *  *
%C A327332                   * *             * *
%C A327332                   **               **
%C A327332                   * * * * * * * * * *
%C A327332 .
%C A327332 Every triangle has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.
%C A327332 For the construction of the sequence the rules are as follows:
%C A327332 On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
%C A327332 At stage 1 we place an V-toothpick, formed of two single toothpicks, with its central vertice directed up, like a gable roof, so a(1) = 1.
%C A327332 For the next n generation we have that:
%C A327332 If n is even then at every free end of the structure we add a I-toothpick formed of two single toothpicks in vertical position.
%C A327332 If n is odd 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 (see the example).
%C A327332 a(n) gives the total number of V-toothpicks and I-toothpicks in the structure after the n-th stage.
%C A327332 A327333 (the first differences) gives the number of elements added at the n-th stage.
%C A327332 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
%C A327332 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 A327332 The structure is almost identical to the structure of A327330, but a little smaller.
%C A327332 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 A327332 The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
%C A327332 It appears that A327330 shares infinitely many terms with this sequence.
%H A327332 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 A327332 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A327332 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A327332 Conjecture: a(2^k) = A327330(2^k), k >= 0.
%e A327332 Illustration of initial terms:
%e A327332 .
%e A327332 .             /\     |/\|
%e A327332 .                    |  |
%e A327332 .
%e A327332 n:     0       1       2
%e A327332 a(n):  0       1       3
%e A327332 After two generations there are only one V-toothpick and two I-toothpicks in the structure, so a(2) = 1 + 2 = 3 (note that in total there are 2*a(2)= 2*3 = 6 single toothpicks of length 1).
%Y A327332 Cf. A139250 (normal toothpicks), A160164 (I-toothpicks), A160722 (a concave pentagon with triangular cells), A161206 (V-toothpicks), A296612, A323641, A323642, A327333 (first differences), A327330 (another version).
%Y A327332 For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770, A323646, A323650.
%K A327332 nonn
%O A327332 0,3
%A A327332 _Omar E. Pol_, Sep 01 2019