cp's OEIS Frontend

From Omar E. Pol, Feb 06 2023: (Start)

The "word" of this cellular automaton is "ab".

Apart from the initial zero 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;

...

Columns "a" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only toothpicks (of length 1).

Columns "b" contain numbers of toothpicks and D-toothpicks when in the top border of the structure there are only D-toothpicks (of length sqrt(2)).

An associated sound to the animation could be (tick, tock), (tick, tock), ..., the same as the ticking clock sound.

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

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

It appears that the right border of the irregular triangle gives the even powers of 2. (End)

This is a cellular automaton of forking paths to 135 degrees which uses elements of two sizes: toothpicks of length 1 and D-toothpicks of length 2^(1/2). Toothpicks are placed in horizontal or vertical direction. D-toothpicks are placed in diagonal direction. Toothpicks and D-toothpicks are connected by their endpoints.

On the infinite square grid we start with no elements.

At stage 1, we place anywhere a D-toothpick.

The rule for adding new elements is as follows. Each exposed endpoint of the elements of the old generation must be touched by the two endpoints of two elements of the new generation such that the angle between the old element and each new element is equal to 135 degrees. Intersections and overlapping are prohibited, so some toothpick endpoints can remain exposed forever.

The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194271) give the number of toothpicks and D-toothpicks added at n-th stage.

If n >> 1 the structure looks like an almost regular octagon. The structure has a fractal-like behavior related to powers of 2 (see formula section in A194271 and A194443). Note that for some values of n we can see an internal growth, similar to A160120. Also there are hidden substructures which have a surprising connection with the Sierpinski triangle. The hidden substructures are displayed more clearly for large values of n without reducing the scale of the drawing. The main "wedges" in the structures are essentially the triangles A194440 and A194442.

Note that this structure is much more complex than A139250 and A160120. The structure contains a large number of distinct polygonal shapes. There are convex polygons and concave polygons, also there are symmetrical and asymmetrical polygons. Several of these polygons are also in the structure of A172310. The number of edges of the known polygons are 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 20, 24. Is not known how many distinct types of polygons there are in the structure if n -> infinite. The sequences related with these polygons are A194276, A194277, A194278 and A194283. Note that the structure is not centered with respect to the axes X, Y. Also, for some polygons the area is not an integer. For symmetric versions of C. A. see A194432 and A194434.

Another representation (Large version): instead toothpicks of length 1 we place toothpicks of length 2. We start with no toothpicks. At stage 1, we place a toothpick of length 2 on the y-axis and centered at the origin. At stage 2 we place four D-toothpicks of length 2^(1/2) = sqrt(2), and so on. In this case the structure is centered with respect to the axes X, Y and the area of the polygons is an integer.

[It appears that a normal toothpick is a line segment of length 1 that is parallel to the x-axis or the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. - N. J. A. Sloane, Feb 06 2023]

Rules:

- Each new toothpick must lie on the hexagonal net such that the toothpick endpoints coincide with two consecutive nodes.

- Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.

The sequence gives the number of toothpicks after n stages. A182841 (the first differences) gives the number added at the n-th stage.

The toothpick structure has polygons in which there are uncovered grid points, the same as A160120 and A161206. For more information see A139250.

Has a behavior similar to A151723, A182632. - Omar E. Pol, Feb 28 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.

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.

The structure has internal growth.

For another version starting from a node see A182632.

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

This cellular automaton has essentially the same rules as A194270. We start at stage 0 with no toothpicks. At stage 1, we place a D-toothpick of length sqrt(2), in diagonal direction, at (0,0),(1,1). At stage 2, we place two toothpicks of length 1. At stage 3 we place four D-toothpicks. And so on. The toothpicks and D-toothpicks are connected by their endpoints. The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A194445) give the number of toothpicks or D-toothpicks added at n-th stage. It appears that the structure shows a fractal (or fractal-like) behavior.

First differs from A220524 at a(13). - Omar E. Pol, Mar 23 2013

The sequence gives the number of V-toothpicks after n-th stages on hexagonal net. A161421 (the first differences) gives the number added at the n-th stage. See A161206 for more information.

Also, it appears this is a H-toothpick sequence in the first quadrant on the square grid, starting with a D-toothpick from the point (0,0). The sequence gives the number of toothpicks and D-toothpicks after n-th stage. A161421 (the first differences) gives the number added at the n-th stage. For more information see A182838.

Number of V-toothpicks added at the n-th stage to the V-toothpick structure of A161420. See also A161206 and A161207.

Also it appears a(n) is also the number of toothpicks and D-toothpicks added at n-th stage to the H-toothpick structure of A161420. See also A182838 and A182839.

We build up a planar graph with hexagonal cells, based on the square grid. There are six kinds of edges.

A U edge is drawn from (x,y) to (x,y+1);

a U^{-1} edge is drawn from (x,y) to (x,y-1);

an L edge is drawn from (x,y) to (x-1,y+1);

an L^{-1} edge is drawn from (x,y) to (x+1,y-1);

an R edge is drawn from (x,y) to (x+1,y+1); and

an R^{-1} edge is drawn from (x,y) to (x-1,y-1).

The construction starts in generation 0 with a single node at the origin (see illustration). At generation 1 we draw a U line from the origin to (0,1).

The graph is then extended using the following rules.

Every U is followed by a pair of lines, L and R;

every L is followed by a U;

every L^{-1} is followed by a pair U^{-1} and R; and

every R is followed by a pair U and L^{-1}.

Lines that fall outside the first quadrant are ignored, and lines that would coincide with existing lines are ignored.

Lines of type U^{-1} and R^{-1} do not need to be followed by anything.

The node numbers in the illustration indicate at which generation the node is reached. This is also the graph distance from the origin.

The number of nodes that are added at the n-th generation, for n >= 0, is given by 1, 1, 1, 2, 4, 4, 5, 5, 7, 7, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 16, 16, 17, 17, 19, 19, ..., with G.f. = (-x^7+x^6+x^4+x^3+1)/((1-x)*(1-x^4)). This is essentially A265428.

The total number of nodes after the n-th generation, for n >= 0, is 1, 2, 3, 5, 9, 13, 18, 23, 30, 37, 45, 53, 63, 73, 84, 95, 108, 121, 135, 149, 165, 181, 198, ... This is essentially A265429.

The number of hexagons that are added at the n-th generation, for n >= 0, is given by 0, 0, 0, 0, 0, 1, 1, 2, 1, 3, 2, 4, 2, 5, 3, 6, 3, 7, 4, 8, 4, 9, 5, 10, 5, 11, 6, 12, 6, 13, ..., with G.f. = x^5*(1+x+x^2)/((1-x^2)*(1-x^4)). This is essentially A106466.