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The word of this cellular automaton is "ab".

The structure of the irregular triangle is as shown below:

a,b;

a,b;

a,b,a,b;

a,b,a,b,a,b,a,b;

a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;

...

Row lengths are the terms of A011782 multiplied by 2, also the column 2 of A296612.

Columns "a" contain numbers of I-toothpicks.

Columns "b" contain numbers of V-toothpicks.

For further information about the word of cellular automata see A296612.

This arises from a hybrid cellular automaton formed of toothpicks of length 2 and D-toothpicks of length 2*sqrt(2).

For the construction of the sequence the rules are as follows:

On the infinite square grid at stage 0 there are no toothpicks, so a(0) = 0.

For the next n generations we have that:

At stage 1 we place a toothpick of length 2 in the horizontal direction, centered at [0,0], so a(1) = 1.

If n is even we add D-toothpicks. Each new D-toothpick must have its midpoint touching the endpoint of exactly one existing toothpick.

If the x-coordinate of the middle point of the D-toothpick is negative then the D-toothpick must be placed in the NE-SW direction.

If the x-coordinate of the middle point of the D-toothpick is positive then the D-toothpick must be placed in the NW-SE direction.

If n is odd we add toothpicks in horizontal direction. Each new toothpick must have its midpoint touching the endpoint of exactly one existing D-toothpick.

The sequence gives the number of toothpicks and D-toothpicks after n stages.

A323647 (the first differences) gives the number of elements added at the n-th stage.

Note that if n >> 1 at the end of every cycle the structure looks like a "volcano", or in other words, the structure looks like a trapeze which is almost an isosceles right triangle.

The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

Another version and very similar to A327330.

This arises from a hybrid cellular automaton on a triangular grid formed of V-toothpicks (A161206) and I-toothpicks (A160164).

After 2^k stages, the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two right triangles (R1 and R2) both with their hypotenuses in vertical position, as shown below:

.

* *

* * * *

* * * *

* * *

* R1 * * R2 *

* * * *

* * * *

* * * *

* * E * *

* * * *

* * * *

** **

* * * * * * * * * *

.

Every triangle has a slight resemblance to Sierpinsky's triangle, but here the structure is much more complex.

For the construction of the sequence the rules are as follows:

On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.

At stage 1 we place an V-toothpick, formed of two single toothpicks, with its central vertice directed up, like a gable roof, so a(1) = 1.

For the next n generation we have that:

If n is even then at every free end of the structure we add a I-toothpick formed of two single toothpicks in vertical position.

If n is odd then at every free end of the structure we add a V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof (see the example).

a(n) gives the total number of V-toothpicks and I-toothpicks in the structure after the n-th stage.

A327333 (the first differences) gives the number of elements added at the n-th stage.

2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.

The structure contains many kinds of polygonal regions, for example: triangles, trapezes, parallelograms, regular hexagons, concave hexagons, concave decagons, concave 12-gons, concave 18-gons, concave 20-gons, and other polygons.

The structure is almost identical to the structure of A327330, but a little smaller.

The behavior seems to suggest that this sequence can be calculated with a formula, in the same way as A139250, but that is only a conjecture.

The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.

It appears that A327330 shares infinitely many terms with this sequence.

An idea from Jean Hoffmann.

In this cellular automaton a V-toothpick is formed by 2 toothpicks of length 1 that share a vertex and the angle between both toothpicks is 60 degrees.

On the infinite triangular grid we start with no V-toothpick, so a(0) = 0.

At stage 1 we place a V-toothpick upside down, so a(1) = 1.

At every stage the V-toothpicks of the new generation must be connected to the structure by touching with their middle vertex the free ends of the V-toothpicks of the previous generation following a special rule:

The new V-toothpicks must be placed between the imaginary straight line containing the two extreme ends of the V-toothpick of the previous generation and the imaginary straight line that contains the middle vertex of that V-toothpick and that it is parallel to the aforementioned straight line.

A355311(n) gives the number of V-toothpicks added to the structure at the n-th stage.

2*a(n) is the total number of toothpicks of length 1 in the structure after n-th stage.

This cellular automaton is a companion of the Y-toothpick cellular automaton of A160120 in the sense that both essentially grow as an equilateral triangle.

This cellular automaton is slightly less symmetrical than Y-toothpick cellular automaton because its structure has a "backbone" formed by concave hexagons from the center of the triangle to one of its vertices.

The behavior could be very close to A160120 and similar to A153006 (see the graph).

After 18 stages we can see in the structure the following polygons:

- Equilateral triangles of perimeter 3.

- Equilateral triangles of perimeter 6 that contain 4 triangular cells.

- Concave hexagons of perimeter 8 that contain 6 triangular cells.

- Concave dodecagons (or concave 12-gons) of perimeter 18 that contain 22 triangular cells.