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This is a cellular automaton of forking paths to 135 degrees which uses elements of three sizes: toothpicks of length 1, D-toothpicks of length 2^(1/2) and D-toothpicks of length 2^(1/2)/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, place a single toothpick on the paper, aligned with the y-axis. 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.

The sequence gives the number of toothpicks and D-toothpicks in the structure after n-th stage. The first differences (A220501) give the number of toothpicks or D-toothpicks added at n-th stage.

It appears that if n >> 1 the structure looks like an octagon. This C.A. has a fractal (or fractal-like) behavior related to powers of 2. Note that for some values of n we can see an internal growth.

The structure contains eight wedges. Each vertical wedge (see A220520) also contains infinitely many copies of the oblique wedges. Each oblique wedge (see A220522) also contains infinitely many copies of the vertical wedges. Finally, each horizontal wedge also contains infinitely many copies of the vertical wedges and of the oblique wedges.

The structure is mysterious: it contains at least 59 distinct internal regions (or polygonal pieces), for example: one of the concave octagons appears for first time at stage 223. The largest known polygon is a concave 24-gon. The exact number of distinct polygons is unknown.

Also the structure contains infinitely many copies of two subsets of distinct size which are formed by five polygons: three hexagons, a 9-gon and a pentagon. These subsets have a surprising connection with the Sierpinski triangle A047999, but the pattern is more complex.

Apparently this cellular automaton has the most complex structure of all the toothpick structures that have been studied (see illustrationsm also the illustrations of the wedges in the entries A220520 and A220522).

The structure contains at least 69 distinct polygonal pieces. The largest known polygon is a concave 24-gon of area 95/2 = 47.5 which appears for first time at stage 879. - Omar E. Pol, Feb 10 2018

From Omar E. Pol, Apr 26 2020: (Start)

The cellular automaton described in A220520 has word "ab", so the structure of this triangle is as follows:

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;

...

The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...

This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks of length 1, and the even-indexed columns (b) contain numbers of the D-toothpicks.

For further information about the "word" of a cellular automaton see A296612. (End)

For n = 0..6 the sequence is similar to some toothpick sequences.

The surprising fact is that for n >= 7 the sequence has periodic tail. More precisely, it has period 3: repeat [8, 16, 12]. This tail is in accordance with the expansion of the four arms of the cross.

This is essentially the first differences of A290221. The behavior is similar to A289841 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

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 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.

Essentially the first differences of A233780.

First differs from A170905 at a(24).

First differs from A233971 at a(25).

First differs from A233765 at a(44).

Essentially the first differences of A220524.

First differs from A194445 at a(13).

Essentially the first differences of A220526.

Essentially the first differences of A220514.

First differs from A194435 at a(13).