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For n = 0..17 the sequence is similar to the known toothpick sequences.

The surprising fact is that for n >= 18 the sequence has a periodic tail. More precisely, it has period 8: repeat [32, 80, 16, 16, 16, 64, 48, 48]. This tail is in accordance with the expansion of the four arms of the cross. The tail also can be written starting from the 20th stage, with period 8: repeat [16, 16, 16, 64, 48, 48, 32, 80], (see example).

This sequence is essentially the first differences of A289840. The behavior is similar to A290221 and A294021 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence. - Omar E. Pol, Oct 29 2017

The sequence arises from a "hybrid" cellular automaton, which consist of two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.

The rules are the same as the rules of A290220 except that here the first element is only one D-toothpick, not two. The result is that here the structure has only two arms instead of four arms as in A290220. On the other hand the structure of each arm is more complex than the structure of the arms of A290220.

The behavior is similar to A289840 and A290220 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. For example: if here the first D-toothpick is placed in the NE-SW direction then the structure grows only in two opposite directions: NW and SE.

On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.

For the next stages we use the following rules:

1) If n is a number of the form 3*k + 1 (cf. A016777), for example: 1, 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected by their endpoints, in the same way as in the odd-indexed stages of A194270.

2) If n is a number of the form 3*k + 2 (cf. A016789), for example: 2, 5, 8, 11, 14, ..., the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the even-indexed stages of A194270.

3) If n is a positive multiple of 3 (cf. A008585), for example: 3, 6, 9, 12, 15, ..., the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.

The minimum width of the structure is 3*2^(1/2) = sqrt(18), see A010474.

The maximum width of the structure is 7*2^(1/2) = sqrt(98), see A010549.

The structure contains seven distinct polygons.

a(n) is the total number of elements in the structure after n generations.

A294021 (the first differences) gives the number of elements added at n-th stage.

The sequence arises from a "hybrid" cellular automaton, which consist essentially in two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.

On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.

For the next stages we have the following rules:

1) At stage 1 we place two D-toothpicks connected by their endpoints on the same diagonal.

2) If n is a number of the form 3*k + 2 (cf. A016789), for example: 2, 5, 8, 11, 14, ..., the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the even-indexed stages of A194270.

3) If n is a positive multiple of 3 (cf. A008585) the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.

4) If n is a number of the form 3*k + 1 (cf. A016777) and > 1, for example: 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected to the structure by their endpoints, in the same way as in the odd-indexed stages of A194270.

a(n) is the total number of elements in the structure after n generations.

A290221 (the first differences) gives the number of elements added at n-th stage.

The surprising fact is that from n = 7 up to 9 the structure is gradually transformed into a square cross.

For n => 9 the shape of the square cross remains forever because its four arms grow indefinitely in the directions North, East, West and South.

Every arm has a width equal to 4.

Every arm also has a periodic structure which can be dissected in infinitely many clusters.

In total, the narrow cross contains five distinct shapes of polygonal regions. There are three polygonal shapes that have an infinite number of copies. On the other hand, two polygonal shapes have a finite number of copies because they are in the center of the cross only. they are the heptagon and the hexagon of area 5.

The structure looks like a square cross but it's simpler than the structure of the complex cross described in A289840.

The behavior is similar to A289840 and A294020 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. - Omar E. Pol, Oct 29 2017

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

It seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.

After 10th stage there are no exposed endpoints in the structure, so the structure is finished.

A294963(n) gives the number of elements added to the structure at n-th stage.

We are on the infinite triangular grid.

At stage 0 there are no elements in the structure, so a(0) = 0.

If n is odd at n-th stage we add Y-toothpicks to the structure.

If n is a positive even number at n-th stage we add V-toothpicks to the structure.

a(n) is the total number of Y-toothpicks and V-toothpicks after n-th stages.

A294981(n) gives the number of elements added to the structure at n-th stage.

The "word" of this cellular automaton is "ab". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

The structure is essentially the same as the finite structure described in A294962 but here there are no D-toothpicks of length sqrt(2)/2. All D-toothpicks in the structure have length sqrt(2).

The same as A294962, it seems that this cellular automaton resembles the synthesis of a molecule, a protein, etc.

After 10th stage there are no exposed endpoints (or free ends), so the structure is finished.

A299771(n) gives the number of elements added to the structure at n-th stage.

The "word" of this cellular automaton is "abcd". For further information about the word of cellular automata see A296612. - Omar E. Pol, Mar 05 2019

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.

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.

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.