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Analog of A151723 and A151725, but here we are working on the hexagonal net where each triangular cell has three neighbors (meeting along its edges). A cell is turned ON if exactly one of its three neighbors is ON. An ON cell remains ON forever.

We start with a single ON cell.

There is a dual version where the triangular cells meet vertex-to-vertex. The counts are the same: the two versions are isomorphic. Reed (1974) uses the vertex-to-vertex version. See the two Sloane "Illustration" links below to compare the two versions.

It appears that a(n) is also the number of polytoothpicks added in a toothpick structure formed by V-toothpicks but starting with a Y-toothpick: a(n) = a(n-1)+(A182632(n)-A182632(n-1))/2. (Checked up to n=39.) - Omar E. Pol, Dec 07 2010 and R. J. Mathar, Dec 17 2010

It appears that the behavior is similar to A161206. - Omar E. Pol, Jan 15 2016

It would be nice to have a formula or recurrence.

If new triangles are required to always move outwards we get A295559 and A295560.

From Paul Cousin, May 23 2025: (Start)

This is ETA rule 242 (11110010 in binary):

-----------------------------------------------

|state of the cell |1|1|1|1|0|0|0|0|

|sum of the neighbors' states |3|2|1|0|3|2|1|0|

|cell's next state |1|1|1|1|0|0|1|0|

----------------------------------------------- (End)

A connected network of toothpicks is constructed by the following iterative procedure. At stage 1, place three toothpicks each of length 1 on a hexagonal net, as a propeller, joined at a node. At each subsequent stage, add two toothpicks (which could be called a single V-toothpick with a 120-degree corner) adjacent to each node which is the endpoint of a single toothpick.

The exposed endpoints of the toothpicks of the old generation are touched by the endpoints of the toothpicks of the new generation. In the graph, the edges of the hexagons become edges of the graph, and the graph grows such that the nodes that were 1-connected in the old generation are 3-connected in the new generation.

It turns out heuristically that this growth does not show frustration, i.e., a free edge is never claimed by two adjacent exposed endpoints at the same stage; the rule of growing the network does apparently not need specifications to address such cases.

The sequence gives the number of toothpicks in the toothpick structure after n-th stage. A182633 (the first differences) gives the number of toothpicks added at n-th stage.

a(n) is also the number of components after n-th stage in a toothpick structure starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >= 2. For example: consider that in A161644 a V-toothpick is also a polytoothpick with two components or toothpicks and a Y-toothpick is also a polytoothpick with three components or toothpicks. For more information about this comment see A161206, A160120 and A161644.

Has a behavior similar to A151723, A182840. - Omar E. Pol, Mar 07 2013

From Omar E. Pol, Feb 17 2023: (Start)

Assume that every triangular cell has area 1.

It appears that the structure contains only three types of polygons:

- Regular hexagons of area 6.

- Concave decagons (or concave 10-gons) of area 12.

- Concave dodecagons (or concave 12-gons) of area 18.

There are infinitely many of these polygons.

The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.

The structure has internal growth.

For an animation see the movie version in the Links section.

The animation shows the fractal-like behavior the same as in other members of the family of toothpick cellular automata.

For another version starting with a simple toothpick see A182840.

For a version of the structure in the first quadrant but on the square grid see A182838. (End)

First differences of A182632.

a(n) is also the number of components added at n-th stage in the toothpick structure formed by V-toothpicks with an initial Y-toothpick, since a V-toothpick has two components and a Y-toothpick has three components (For more information see A161206, A160120, A161644).

See the comments in A161644.

It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - Omar E. Pol, Dec 07 2010

An H-toothpick sequence is a toothpick sequence on a square grid that resembles a partial honeycomb of hexagons.

The structure has two types of elements: the classic toothpicks with length 1 and the "D-toothpicks" with length sqrt(2).

Classic toothpicks are placed in the vertical direction and D-toothpicks are placed in a diagonal direction.

Each hexagon has area = 4.

The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.

The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first differences) gives the number added at the n-th stage.

[It appears that for this sequence a classic toothpick is a line segment of length 1 that is parallel to the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. It also appears that classic toothpicks are not placed on the y-axis. - N. J. A. Sloane, Feb 06 2023]

From Omar E. Pol, Feb 17 2023: (Start)

This cellular automaton appears to be a version on the square grid of the first quadrant of the structure of A182840.

The rules are as follows:

- The elements (toothpicks and D-toothpicks) are connected at their ends.

- At each free end of the elements of the old generation two elements of the new generation must be connected.

- The toothpicks of length 1 must always be placed vertically, i.e. parallel to the Y-axis.

- The angle between a toothpick of length 1 and a D-toothpick of length sqrt(2) that share the same node must be 135 degrees, therefore the angle between two D-toothpicks that share the same node is 90 degrees.

As a result of these rules we can see that in the odd-indexed rows of the structure are placed only the toothpicks of length 1 and in the even-indexed rows of the structure are placed the D-toothpicks of length sqrt(2).

Apart from the trapezoids, pentagons and heptagons that are adjacent to the axes of the first quadrant it appears that there are only three types of polygons:

- Regular hexagons of area 4.

- Concave decagons (or concave 10-gons) of area 8.

- Concave dodecagons (or concave 12-gons) of area 12.

There are infinitely many of these polygons.

The structure shows a fractal-like behavior as we can see in other members of the family of toothpick cellular automata.

The structure has internal growth as some members of the mentioned family. (End)

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks and V-toothpicks. Also, it appears that this is a missing link between A147562 (Ulam-Warburton) and three toothpick sequences: A139250 (normal toothpicks), A161206 (V-toothpicks) and A160120 (Y-toothpicks). The behavior resembles the toothpick sequence A139250, on the other hand, the formulas are directly related to A147562. Plot 2 shows that the graph is located between the graph of A139250 and the graph of A147562.

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 such that the angle of each single toothpick with respect to the connected I-toothpick is 120 degrees.

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.

A323651 (the first differences) gives the number of elements added at the n-th stage.

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

The structure contains only three kinds of polygonal regions as follows:

- Rhombuses that contain two triangular cells.

- Regular hexagons that contain six triangular cells.

- Oblong hexagons that contain 10 triangular cells.

The structure looks like a "garden of flowers with six petals" (between other substructures). In particular, after 2^(n+1) stages with n >= 0, the structure looks like a flower garden in a rectangular box which contains A002450(n) flowers with six petals.

Note that this hybrid cellular automaton is also a superstructure of the Ulam-Warburton cellular automaton (at least in four ways). The explanation is as follows:

1) A147562(n) equals the total number of I-toothpicks in the structure after 2*n - 1 stage, n >= 1.

2) A147562(n) equals the total number of pairs of Y-toothpicks connected by their endpoints in the structure after 2*n stage (see the example).

3) A147562(n) equals the total number of "flowers with six petals" (or six-pointed stars formed of six rhombuses) in the structure after 4*n stage. Note that the location of the "flowers with six petals" in the structure is essentially the same as the location of the "ON" cells in the version "one-step bishop" of A147562.

4) For more connections to A147562 see the Formula section.

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

The total number of “flowers with six petals” after n-th stage equals the total number of “hidden crosses” after n-th stage in the toothpick structure of A139250, including the central cross (beginning to count the crosses when their “nuclei” are totally formed with 4 quadrilaterals). - Omar E. Pol, Mar 06 2019

The same as A172310 but starting with two L-toothpicks.

We start at stage 0 with no L-toothpicks.

At stage 1 we place two large L-toothpicks in the horizontal direction, as a "X", anywhere in the plane.

At stage 2 we place four small L-toothpicks.

At stage 3 we add eight more large L-toothpicks.

At stage 4 we add eight more small L-toothpicks.

And so on ...

The L-toothpick cellular automaton has an unusual property: the growths in its four wide wedges [North, East, South and West] have a recurrent behavior related to powers of 2, as we can find in other cellular automata (i.e., A212008). On the other hand, in its four narrow wedges [NE, SE, SW, NW] the behavior seems to be chaotic, without any recurrence, similar to the behavior of the snowflake cellular automaton of A161330. The remarkable fact is that with the same rules, different behaviors are produced. (See Applegate's movie version in the Links section.) - Omar E. Pol, Nov 06 2018

The sequence gives the number of toothpicks after n stages. A182635 (the first differences) gives the number added at the n-th stage.

The 120-degree wedge defines a conic region which toothpicks (except one end point of the initial toothpick) are not allowed to cross or touch. The wings of the wedge point +-60 degrees away from the pointing direction of the initial toothpick.

Toothpicks are connected by their endpoints, the same as the toothpicks of A182632.

First differs from A139250 at a(11).

The odd-indexed terms (a bisection) gives A147582, the first differences of A147562 (Ulam-Warburton cellular automaton).

The even-indexed terms (a bisection) gives A147582 multiplied by 2.

The word of this cellular automaton is "ab", so 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. See the example.

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

The same as A172310 and A172304, but starting from half L-toothpick in the first quadrant.

Note that if n is odd then we add the small L-toothpicks to the structure, otherwise we add the large L-toothpicks to the structure.

We start at stage 0 with half L-toothpick: A segment from (0,0) to (1,1).

At stage 1 we place a small L-toothpick at the exposed toothpick end.

At stage 2 we place two large L-toothpicks.

At stage 3 we place two small L-toothpicks.

At stage 4 we place two large L-toothpicks.

And so on...

The sequence gives the number of L-toothpicks after n stages. A172309 (the first differences) gives the number of L-toothpicks added at the n-th stage.