| A Y-toothpick is a "toothpick" with three endpoints, formed from three half-toothpicks, like a propeller.
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 place three other Y-toothpicks. After round 2, in the sieve we can see three rhombuses and a hexagon (see illustrations).
At round 3 we place three other Y-toothpicks.
And so on...
The sequence gives the number of Y-toothpicks after n rounds. A160121 (the first differences) gives the number added at the n-th round.
It appears that the Y-toothpick pattern has a recursive, fractal-like structure. An animation can show the like-fractal behavior.
See the entry A139250 for more information about the toothpick process and the toothpick propagation.
Note that, on the infinite triangular grid, a Y-toothpick can be represented as a polyedge with three components. In this case, at 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 an external propagation and an internal propagation of the Y-toothpicks.
Also, in this structure we can see distinct polygons, with 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-agons .. (with 7 non covered grid points)
- 30-agons .. (with 26 non covered grid points)
- ...
Observation: It appears that the number of distinct polygons that contains non covered grid points is infinite.
Apparently, this sequence is related to powers of 2, for example:
Conjecture: It appears that if n = 2^k, k>0, then, between the other polygons, 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 A160715 for another version of this structure but without internal growth of Y-toothpicks. [From Omar E. Pol (info(AT)polprimos.com), May 31 2010]
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