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Essentially the first differences of A220500.

The cellular automaton of A220500 contains a large number of distinct polygonal shapes. The exact number is unknown. Apparently it's greater than 63.

For simplicity we also call polygonal shapes "polygons".

In order to construct this sequence we use the following rules:

- Consider only the convex polygons and the concave polygons. Self-intersecting polygons are not counted.

- Unfinished polygons with inward growth are not counted.

- If two polygons have the same shape but they have different size then these polygons must be counted as distinct polygonal shapes.

- The reflected shapes of asymmetric polygons, both with the same area, must be counted as distinct polygonal shapes.

Question: Is there a maximal record in this sequence?

We define an "L-toothpick" to consist of two line segments forming an "L".

There are two size for L-toothpicks: Small and large. Each component of small L-toothpick has length 1. Each component of large L- toothpick has length sqrt(2).

The rule for the n-th stage:

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

Note that, on the infinite square grid, every large L-toothpick is placed with angle = 45 degrees and every small L-toothpick is placed with angle = 90 degrees.

The special rule: L-toothpicks are not added if this would lead to overlap with another L-toothpick branch in the same generation.

We start at stage 0 with no L-toothpicks.

At stage 1 we place a large L-toothpick in the horizontal direction, as a "V", anywhere in the plane (Note that there are two exposed endpoints).

At stage 2 we place two small L-toothpicks.

At stage 3 we place four large L-toothpicks.

At stage 4 we place six small L-toothpicks.

And so on...

The sequence gives the number of L-toothpick after n stages. A172311 (the first differences) gives the number of L-toothpicks added at the n-th stage.

For more information see A139250, the toothpick sequence.

In calculating the extension, the "special rule" was strengthened to prohibit intersections as well as overlappings. [From John W. Layman, Feb 04 2010]

Note that the endpoints of the L-toothpicks of the new generation can touch the L-toothpìcks of old generations but the crosses and overlaps are prohibited. - Omar E. Pol, Mar 26 2016

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., A194270). 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 structure is essentially one of the vertical wedges of several D-toothpick structures. For more information see A220500. First differs from A194440 at a(14). The first differences (A220521) give the number of toothpicks or D-toothpicks added at n-th stage. See A220522 for the "narrow" triangle of the third kind.

The structure is essentially one of the oblique wedges of several D-toothpick structures. For more information see A220500. First differs from A194442 at a(47). The first differences (A220523) give the number of toothpicks or D-toothpicks added at n-th stage.

The sequence arises from a "hybrid" cellular automaton on the infinite square grid, which consist of two successive generations using toothpicks of length 2 (cf. A139250) followed by two successive generations using the rules of the D-toothpick sequence A220500.

In other words (and more precisely) we have that:

1) If n is congruent to 1 or 2 mod 4 (cf. A042963), for example: 1, 2, 5, 6, 9, 10, ..., 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.

2) If n is a positive integer of the form 4*k-1 (cf. A004767), for example: 3, 7, 11, 15, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) and eventually D-toothpicks of length sqrt(2)/2, in both cases the D-toothpicks are connected to the structure by their endpoints, in the same way as in the even-indexed stages of A220500.

3) If n is a positive multiple of 4 (cf. A008586) 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 odd-indexed stages of A220500.

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

A289841 (the first differences) gives the number of elements added at n-th stage.

Note that after 19 generations the structure is a 72-gon which essentially looks like a diamond (as a square that has been rotated 45 degrees).

The surprising fact is that from n = 20 up to 27 the structure is gradually transformed into a square cross.

The diamond mentioned above can be interpreted as the center of the cross. The diamond has an area equal to 384 and it contains 222 polygonal regions (or enclosures) of 11 distinct shapes. Missing two heptagonal shapes which are in the arms of the square cross only.

In total the complex cross contains 13 distinct shapes of polygonal regions. There are ten polygonal shapes that have an infinite number of copies. On the other hand, three of these polygonal shapes have a finite number of copies because they are in the center of the cross only. For example: there are only four copies of the concave 14-gon, which is also the largest polygon in the structure.

For n => 27 the shape of the square cross remains forever because its four arms grow indefinitely.

Every arm has a minimum width equal to 8, and a maximum width equal to 12.

Every arm also has a periodic structure which can be dissected in infinitely many clusters of area equal to 64. Every cluster is a 30-gon that contains 40 polygonal regions of nine distinct shapes.

If n is a number of the form 8*k-3 (cf. A017101) and greater than 19, for example: 27, 35, 43, 51, ..., then at n-th stage a new cluster is finished in every arm of the cross.

The behavior is similar to A290220 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

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

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

The cells are the squares of the standard square grid.

Cells are either OFF or ON, once they are ON they stay ON forever.

Each cell has 8 neighbors, the cells that are a knight's move away.

We begin in generation 1 with a single ON cell.

A cell is turned ON at generation n+1 if it has exactly one ON neighbor at generation n.

(Since cells stay ON, an equivalent definition is that a cell is turned ON at generation n+1 if it has exactly one neighbor that has been turned ON at some earlier generation. - N. J. A. Sloane, Dec 19 2018)

This sequence has similarities with A151725: here we use knight moves, there we use king moves.

This is a knight's-move version of the Ulam-Warburton cellular automaton (see A147562). - N. J. A. Sloane, Dec 21 2018

The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections, which generate the dihedral group D_8 of order 8), so A319019 is a multiple of 8 (compare A322050). - N. J. A. Sloane, Dec 16 2018

From Omar E. Pol, Dec 16 2018: (Start)

For n >> 1 (for example: n = 257) the structure of this sequence is similar to the structure of both A194270 and of A220500, the D-toothpick cellular automata of the second kind and of the third kind respectively. The animations of both CAs are in the Applegate's movie version.

Also, the graph of A319018 is a bit similar to the graph of A245540, which is essentially a 45-degree-3D-wedge of A245542 (a pyramid) which is the partial sums of A160239 (Fredkin's replicator). See "Plot 2": A319018 vs. A245540. (End)

The conjecture that A322050(2^k+1)=1 also suggests a fractal geometry. Let P_k be the associated set of eight points. It appears that P_k may be written as the intersection of four fixed lines, y = +-2*x and x = +-2*y, with a circle, x^2 + y^2 = 5*4^k (see linked image "Log-Periodic Coloring"). - Bradley Klee, Dec 16 2018

In many of these toothpick or cellular automata sequences it is common to see graphs which look like some version of the famous blancmange curve (also known as the Takagi curve). I expect that is what we are seeing when we look at the graph of A322049, although we probably need to go a lot further out before the true shape becomes apparent. - N. J. A. Sloane, Dec 17 2018

The graph of A322049 (related to first differences of this sequence) appears to have rather a self-similar structure which repeats at powers of 2, and more specifically at 2^10 = 1024. There is no central symmetry or continuity, which are characteristic properties of the blancmange curve. - M. F. Hasler, Dec 28 2018

The 8 points added in generation n = 2^k + 1 are P_k = 2^k*K where K = {(+-2, +-1), (+-1, +-2)} is the set of the initial 8 knight moves. So P_k is indeed the intersection of the rays of slope +-1/2 resp. +-2 and a circle of radius 2^k*sqrt(5). In the subsequent generation n = 2^k + 2, the new cells switched on are exactly the 7 "new" knight move neighbors of these 8 cells, (P_k + K) \ (2^k - 1)*K. The 8th neighbor, lying one knight move closer to the origin, has been switched on in generation 2^k, together with an octagonal "wall" consisting of every other cell on horizontal and vertical segments between these points (2^k - 1)*K, and all cells on the diagonal segments between these points, as well as 2 more diagonals just next to these (on the inner side) and shorter by 2 cells (so they are empty for k = 1). This yields 4*(2 + (2^k - 2)*(1+3)) new ON cells in generation 2^k, plus 8*(2^(k-1) - 2) more new ON cells on horizontal, vertical and diagonal lines 4 units closer to the origin for k > 2, and similar additional terms for k > 4 etc. - M. F. Hasler, Dec 28 2018