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A160120 Y-toothpick sequence (see Comments lines for definition). 87
0, 1, 4, 7, 16, 19, 28, 37, 58, 67, 76, 85, 106, 121, 142, 169, 220, 247, 256, 265, 286, 301, 322, 349, 400, 433, 454, 481, 532, 583, 640, 709, 826, 907, 928, 937, 958, 973, 994, 1021, 1072, 1105, 1126, 1153, 1204, 1255, 1312, 1381, 1498, 1585, 1618, 1645 (list; graph; refs; listen; history; text; internal format)



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.

On the infinite triangular grid, we start at round 0 with no Y-toothpicks.

At round 1 we place a Y-toothpick anywhere in the plane.

At round 2 we add three more Y-toothpicks. After round 2, in the structure there are three rhombuses and a hexagon.

At round 3 we add three more Y-toothpicks.

And so on ... (see illustrations).

The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.

The Y-toothpick pattern has a recursive, fractal (or fractal-like) structure.

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.

This structure is more complex than the toothpick structure of A139250. For example, at some rounds we can see inward growth.

The structure contains distinct polygons which have side length equal to 1.

Observation: It appears that the region of the structure where all grid points are covered is formed only by three distinct polygons:

- Triangles

- Rhombuses

- Concave-convex hexagons

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:

- Decagons   (with  1 non-covered grid point)

- Dodecagons (with  4 non-covered grid points)

- 18-gons    (with  7 non-covered grid points)

- 30-gons    (with 26 non-covered grid points)

- ...

Observation: It appears that the number of distinct polygons that contain non-covered grid points is infinite.

This sequence appears to be related to powers of 2. For example:

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.

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".

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.

See the entry A139250 for more information about the growth of "standard" toothpicks.

See also A160715 for another version of this structure but without internal growth of Y-toothpicks. [Omar E. Pol, May 31 2010]

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

The behavior is similar to A153006 (see the graph). - Omar E. Pol, Apr 03 2018


JungHwan Min, Table of n, a(n) for n = 0..5000

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, 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.]

David Applegate, The movie version

Omar E. Pol, Illustration of initial terms [From Omar E. Pol, Jun 01 2009]

Omar E. Pol, Illustration of the structure after 17 stages

Omar E. Pol, Illustration of the structure after 34 stages, from Applegate's movie version.

Omar E. Pol, Illustration: Fractal recursion, general step. (1)

Omar E. Pol, Illustration of initial terms of A139250, A160120, A147562 (Overlapping figures)

Omar E. Pol, Illustration of initial terms of A160120, A161206, A161328, A161330 (Triangular grid and toothpicks)

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to toothpick sequences

Index entries for sequences related to cellular automata


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 *)

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 *)


Toothpick sequence: A139250.

Cf. A000079, A000225, A006516, A147562, A153006, A160121, A160123, A160715, A161206, A161328, A161330, A161430, A173066, A173068, A253770.

Sequence in context: A166700 A266532 A160715 * A130665 A236305 A212062

Adjacent sequences:  A160117 A160118 A160119 * A160121 A160122 A160123




Omar E. Pol, May 02 2009


More terms from David Applegate, Jun 14 2009, Jun 18 2009



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Last modified February 28 09:24 EST 2021. Contains 341695 sequences. (Running on oeis4.)