

A160120


Ytoothpick sequence (see Comments lines for definition).


76



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
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OFFSET

0,3


COMMENTS

A Ytoothpick (or Yshaped toothpick) is formed from three toothpicks of length 1, like a star with three endpoints and only one middlepoint.
On the infinite triangular grid, we start at round 0 with no Ytoothpicks.
At round 1 we place a Ytoothpick anywhere in the plane.
At round 2 we add three more Ytoothpicks. After round 2, in the structure there are three rhombuses and a hexagon.
At round 3 we add three more Ytoothpicks.
And so on ... (see illustrations).
The sequence gives the number of Ytoothpicks after n rounds. A160121 (the first differences) gives the number added at the nth round.
The Ytoothpick pattern has a recursive, fractal (or fractallike) structure.
Note that, on the infinite triangular grid, a Ytoothpick can be represented as a polyedge with three components. In this case, at the nth 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
 Concaveconvex hexagons
Holes in the structure: Also, we can see distinct concaveconvex polygons which contains a region where there are no grid points that are covered, for example:
 Decagons (with 1 noncovered grid point)
 Dodecagons (with 4 noncovered grid points)
 18gons (with 7 noncovered grid points)
 30gons (with 26 noncovered grid points)
 ...
Observation: It appears that the number of distinct polygons that contain noncovered 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 triangleshaped 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 Ytoothpick 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 Ytoothpicks. [Omar E. Pol, May 31 2010]
For an alternative visualization replace every single toothpick with a rhombus, or in other words, replace every Ytoothpick with the "threediamond" symbol, so we have a cellular automaton in which a(n) gives the total number of "threediamond" symbols after nth stage and A160167(n) counts the total number of "ON" diamonds in the structure after nth 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


LINKS

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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)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


MATHEMATICA

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 *)


CROSSREFS

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


KEYWORD

nonn,changed


AUTHOR

Omar E. Pol, May 02 2009


EXTENSIONS

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


STATUS

approved



