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

The surprising fact is that after 2^k stages the structure looks like a concave pentagon, which is formed essentially by an equilateral triangle (E) surrounded by two quadrilaterals (Q1 and Q2), both with their largest sides in vertical position, as shown below:

.

* *

* * * *

* * * *

* * *

* Q1 * Q2 *

* * * *

* * * *

* * * *

* * * *

* * E * *

* * * *

* * * *

** **

* * * * * * * * * *

.

Note that for n >> 1 both quadrilaterals look like right triangles.

Every polygon 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 I-toothpick formed of two single toothpicks in vertical position, 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 V-toothpick, formed of two single toothpicks, with its central vertex directed upward, like a gable roof.

If n is odd then we add I-toothpicks in vertical position (see the example).

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

A327331 (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 A327332, but a little larger at the upper edge.

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.

For another version, very similar, starting with a V-toothpick, see A327332, which it appears that shares infinitely many terms with this sequence.

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.

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.

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 V-toothpicks. Columns "b" contain numbers of I-toothpicks. See the example.

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

The rules are the same as the rules of A296510 (the toothpick sequence on triangular grid) but here we are in a 60-degree wedge. For the position of the initial toothpicks see the example.

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

A296611, the first differences, gives the number of toothpicks added at n-th stage.

The "word" of this cellular automaton is "abc", the same as the word of A296510. For more information about the word of cellular automata see A296612.

The word of this cellular automaton is "abc", the same as the word of A296511, but here the irregular triangle starts with three rows of length 3 as shown below:

a,b,c;

a,b,c;

a,b,c;

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

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

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

...

Row lengths are 3, 3, 3, 6, 12, 24, 48, 96, ... or in other words: 3 together with the column 3 of A296612.

See also the example.

Inspired by Neil Sloane's presentation at Rutgers' Experimental Mathematics Seminar (see the Links section).

Beneath the familiar image of every cellular automaton lies an infinite world of hidden patterns, stained-glass windows, gaskets, curves and fractals.

As an example of this statement we will focus on the "toothpick" cellular automata. In general after 2^k stages, k >= 2, these structures looks like the framework of a stained-glass window (without the colored glass). Toothpicks represent the cames of the structure. Now the idea is to put the stained glass.

Here we will use the "toothpick" cellular automaton of A139250.

After every stage the square cells of the newly formed regions will be colored.

We have two colors. If n is odd, they are painted with the color 1. If n is even, they are painted with the color 2.

Note that there are infinitely many rules for coloring a cellular automaton since there are infinitely many colors related to infinitely many sequences, however, the rule used here seems quite natural, since the number of colors coincides with the number of letters of the "word" of this cellular automaton, which is "ab". So here we have toothpicks on the two axes of the infinite square grid, two associated sounds (tick-tock) and two colors.

After 2^k stages, k >= 2, a rectangular-stained-glass window with two colors will have been formed.

Conjecture 1: after 2^k stages the number of cells of color 1 is equal to the number of cells of color 2.

Conjecture 2: after 2^k stages, k >= 2, in the structure there are essentially one major region of color 1 and two major regions of color 2.

It appears that there are certain sub-quadrants that have the complementary structure and the opposite colors of other sub-quadrants.

This sequence is a square array read by antidiagonals upwards that represents the colors (1 or 2) of every cell in the fourth quadrant of the stained-glass windows. The corner of the array represents the cell whose upper-left vertex is the point (0,0) of the fourth quadrant of the structure.

For a binary sequence the 2's should be replaced with 0's.

Note that for the toothpick cellular automaton on triangular grid of A296510 (whose word is "abc") three colors should be used there. Same for the C.A. of A299476 and of A299478.

For more information on the "word" of a cellular automaton see A296612 and see ALSO the third triangle in the example section of A139251.

The following three steps refer to the visualization of hidden gaskets and hidden curves from the stained-glass windows of the toothpick structures.

First, a growth limit is set until the final stage 2^k.

Then the line segments other than the border between the two colors are removed.

Finally the colors are also removed.

In this case, two curves will be formed. One curve on the first and second quadrant and the other curve on the third and fourth quadrant. One curve is the reflection of the other.

After studying and analyzing the curve, a sequence and an animation could be made to represent it, from the stage 1 to n.

The curve obtained resembles the Hilbert curve and the Moore curve, but apparently here the curve is a bit more complex (see the example).

An idea from Jean Hoffmann.