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A323650 Flower garden sequence (see Comments for precise definition).

%I A323650 #135 Jun 18 2022 18:22:07
%S A323650 0,1,3,7,15,19,27,39,63,67,75,87,111,123,147,183,255,259,267,279,303,
%T A323650 315,339,375,447,459,483,519,591,627,699,807,1023,1027,1035,1047,1071,
%U A323650 1083,1107,1143,1215,1227,1251,1287,1359,1395,1467,1575,1791,1803,1827,1863,1935,1971,2043,2151,2367,2403,2475
%N A323650 Flower garden sequence (see Comments for precise definition).
%C A323650 This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks and V-toothpicks. Also, it appears that this is a missing link between A147562 (Ulam-Warburton) and three toothpick sequences: A139250 (normal toothpicks), A161206 (V-toothpicks) and A160120 (Y-toothpicks). The behavior resembles the toothpick sequence A139250, on the other hand, the formulas are directly related to A147562. Plot 2 shows that the graph is located between the graph of A139250 and the graph of A147562.
%C A323650 For the construction of the sequence the rules are as follows:
%C A323650 On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
%C A323650 At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
%C A323650 For the next n generation we have that:
%C A323650 If n is even then at every free end of the structure we add a V-toothpick formed of two single toothpicks such that the angle of each single toothpick with respect to the connected I-toothpick is 120 degrees.
%C A323650 If n is odd then we add I-toothpicks in vertical position (see the example).
%C A323650 a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
%C A323650 A323651 (the first differences) gives the number of elements added at the n-th stage.
%C A323650 Note that 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
%C A323650 The structure contains only three kinds of polygonal regions as follows:
%C A323650 - Rhombuses that contain two triangular cells.
%C A323650 - Regular hexagons that contain six triangular cells.
%C A323650 - Oblong hexagons that contain 10 triangular cells.
%C A323650 The structure looks like a "garden of flowers with six petals" (between other substructures). In particular, after 2^(n+1) stages with n >= 0, the structure looks like a flower garden in a rectangular box which contains A002450(n) flowers with six petals.
%C A323650 Note that this hybrid cellular automaton is also a superstructure of the Ulam-Warburton cellular automaton (at least in four ways). The explanation is as follows:
%C A323650 1) A147562(n) equals the total number of I-toothpicks in the structure after 2*n - 1 stage, n >= 1.
%C A323650 2) A147562(n) equals the total number of pairs of Y-toothpicks connected by their endpoints in the structure after 2*n stage (see the example).
%C A323650 3) A147562(n) equals the total number of "flowers with six petals" (or six-pointed stars formed of six rhombuses) in the structure after 4*n stage. Note that the location of the "flowers with six petals" in the structure is essentially the same as the location of the "ON" cells in the version "one-step bishop" of A147562.
%C A323650 4) For more connections to A147562 see the Formula section.
%C A323650 The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
%C A323650 The total number of “flowers with six petals” after n-th stage equals the total number of “hidden crosses” after n-th stage in the toothpick structure of A139250, including the central cross (beginning to count the crosses when their “nuclei” are totally formed with 4 quadrilaterals). - _Omar E. Pol_, Mar 06 2019
%H A323650 Paolo Xausa, <a href="/A323650/b323650.txt">Table of n, a(n) for n = 0..9999</a>
%H A323650 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 A323650 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A323650 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A323650 a(n) = 3*A147562(n/2) if n is even.
%F A323650 a(n) = 3*A147562((n-1)/2) + A147582(n) if n is odd.
%F A323650 a(n) = 3*A147562((n-1)/2) + A147562(n) - A147562(n-1) if n is odd.
%F A323650 a(2^n) = A103454(n), n >= 0.
%e A323650 Illustration of initial terms:
%e A323650 .
%e A323650                         |   |
%e A323650                 \ /     |\ /|
%e A323650          |       |        |
%e A323650          |       |        |
%e A323650                 / \     |/ \|
%e A323650                         |   |
%e A323650 n        1       2        3
%e A323650 a(n)     1       3        7
%e A323650 .
%e A323650 Note that for n = 2 the structure is also the same as a pair of Y-toothpicks connected by their endpoints (see A160120).
%t A323650 bw3[n_]:=bw3[n]=3^(DigitCount[n,2,1]-1);
%t A323650 A147562[n_]:=A147562[n]=If[n<2,n,1+4Sum[bw3[k],{k,n-1}]];
%t A323650 A323650[n_]:=If[OddQ[n],3A147562[(n-1)/2]+A147562[n]-A147562[n-1],3A147562[n/2]];
%t A323650 nterms=100;Array[A323650,nterms,0] (* _Paolo Xausa_, Jun 17 2022 *)
%Y A323650 Cf. A002450, A103454, A139250 (normal toothpicks), A147562 (Ulam-Warburton), A147582, A160120 (Y-toothpicks), A161206 (V-toothpicks), A296612, A323641, A323642, A323649 (corner sequence), A323651 (first differences).
%Y A323650 For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770.
%K A323650 nonn
%O A323650 0,3
%A A323650 _Omar E. Pol_, Jan 21 2019