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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

Essentially the first differences of A294020.

The sequence starts with 0, 1, 4, 4, 6. For n >= 5 the sequence has a periodic tail. More precisely, it has period 6: repeat [8, 4, 14, 24, 16, 22]. This tail is in accordance with the expansion of the two arms of the structure.

The behavior is similar to A289841 and A290221 in the sense that these three sequences from cellular automata have the property that after the initial terms the continuation is a periodic sequence.

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.

The word of this cellular automaton is abcd. For more information see A296612.

The "word" of this cellular automaton is "ab", but note that this triangle has an unusual structure: an additional row of length 2. For more information about the word of cellular automata see A296612.

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,a,b;

...

Row lengths give 2 together with the terms of A011782 multiplied by 2, also 2 togheter with the column 2 of A296612.

Columns "a" contain numbers of toothpicks of length 2.

Columns "b" contain numbers of D-toothpicks of length 2*sqrt(2). See the example.

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.

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.