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A160120 Y-toothpick sequence (see Comments lines for definition).

%I A160120 #80 Dec 18 2024 14:59:12
%S A160120 0,1,4,7,16,19,28,37,58,67,76,85,106,121,142,169,220,247,256,265,286,
%T A160120 301,322,349,400,433,454,481,532,583,640,709,826,907,928,937,958,973,
%U A160120 994,1021,1072,1105,1126,1153,1204,1255,1312,1381,1498,1585,1618,1645
%N A160120 Y-toothpick sequence (see Comments lines for definition).
%C A160120 A Y-toothpick (or Y-shaped toothpick) is formed from three toothpicks of length 1, like a star with three endpoints and only one middle-point.
%C A160120 On the infinite triangular grid, we start at round 0 with no Y-toothpicks.
%C A160120 At round 1 we place a Y-toothpick anywhere in the plane.
%C A160120 At round 2 we add three more Y-toothpicks. After round 2, in the structure there are three rhombuses and a hexagon.
%C A160120 At round 3 we add three more Y-toothpicks.
%C A160120 And so on ... (see illustrations).
%C A160120 The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.
%C A160120 The Y-toothpick pattern has a recursive, fractal (or fractal-like) structure.
%C A160120 Note that, on the infinite triangular grid, a Y-toothpick can be represented as a polyedge with three components. In this case, at the n-th round, the structure is a polyedge with 3*a(n) components.
%C A160120 This structure is more complex than the toothpick structure of A139250. For example, at some rounds we can see inward growth.
%C A160120 The structure contains distinct polygons which have side length equal to 1.
%C A160120 Observation: It appears that the region of the structure where all grid points are covered is formed only by three distinct polygons:
%C A160120 - Triangles
%C A160120 - Rhombuses
%C A160120 - Concave-convex hexagons
%C A160120 Holes in the structure: Also, we can see distinct concave-convex polygons which contains a region where there are no grid points that are covered, for example:
%C A160120 - Decagons   (with  1 non-covered grid point)
%C A160120 - Dodecagons (with  4 non-covered grid points)
%C A160120 - 18-gons    (with  7 non-covered grid points)
%C A160120 - 30-gons    (with 26 non-covered grid points)
%C A160120 - ...
%C A160120 Observation: It appears that the number of distinct polygons that contain non-covered grid points is infinite.
%C A160120 This sequence appears to be related to powers of 2. For example:
%C A160120 Conjecture: It appears that if n = 2^k, k>0, then, between the other polygons, there appears a new centered hexagon formed by three rhombuses with side length = 2^k/2 = n/2.
%C A160120 Conjecture: Consider the perimeter of the structure. It appears that if n = 2^k, k>0, then the structure is a triangle-shaped polygon with A000225(k)*6 sides and a half toothpick in each vertice of the "triangle".
%C A160120 Conjecture: It appears that if n = 2^k, k>0, then the ratio of areas between the Y-toothpick structure and the unitary triangle is equal to A006516(k)*6.
%C A160120 See the entry A139250 for more information about the growth of "standard" toothpicks.
%C A160120 See also A160715 for another version of this structure but without internal growth of Y-toothpicks. [_Omar E. Pol_, May 31 2010]
%C A160120 For an alternative visualization replace every single toothpick with a rhombus, or in other words, replace every Y-toothpick with the "three-diamond" symbol, so we have a cellular automaton in which a(n) gives the total number of "three-diamond" symbols after n-th stage and A160167(n) counts the total number of "ON" diamonds in the structure after n-th stage. See also A253770. - _Omar E. Pol_, Dec 24 2015
%C A160120 The behavior is similar to A153006 (see the graph). - _Omar E. Pol_, Apr 03 2018
%H A160120 JungHwan Min, <a href="/A160120/b160120.txt">Table of n, a(n) for n = 0..5000</a>
%H A160120 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 A160120 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A160120 Ayliean McDonald, <a href="https://twitter.com/ayliean/status/1363175938796773377?lang=bg">Y-toothpick fractal</a>, Twitter video (2021).
%H A160120 Olbaid Fractalium, <a href="https://www.youtube.com/watch?v=jledHAioyJo">Toothpick sequence Part 2</a>, Youtube video (2023).
%H A160120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltpy58.jpg"> Illustration of initial terms</a>
%H A160120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltpy17.jpg">Illustration of the structure after 17 stages</a>
%H A160120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltpy34.jpg">Illustration of the structure after 34 stages</a>, from Applegate's movie version.
%H A160120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltpyf1.jpg">Illustration: Fractal recursion, general step. (1)</a>
%H A160120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca001.jpg">Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)</a>
%H A160120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poltp120.jpg">Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks)</a>
%H A160120 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 A160120 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H A160120 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%t A160120 YTPFunc[lis_, step_] := With[{out = Extract[lis, {{1, 2}, {2, 1}, {-1, -1}}], in = lis[[2, 2]]}, Which[in == 0 && Count[out, 2] >= 2, 1, in == 0 && Count[out, 2] == 1, 2, True, in]]; A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{YTPFunc, {}, {1, 1}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* _JungHwan Min_, Jan 28 2016 *)
%t A160120 A160120[0] = 0; A160120[n_] := With[{m = n - 1}, Count[CellularAutomaton[{435225738745686506433286166261571728070, 3, {{-1, 0}, {0, -1}, {0, 0}, {1, 1}}}, {{{2}}, 0}, {{{m}}}], 2, 2]] (* _JungHwan Min_, Jan 28 2016 *)
%Y A160120 Toothpick sequence: A139250.
%Y A160120 Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.
%K A160120 nonn
%O A160120 0,3
%A A160120 _Omar E. Pol_, May 02 2009
%E A160120 More terms from _David Applegate_, Jun 14 2009, Jun 18 2009