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:
- 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 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. [From Omar E. Pol, May 31 2010]