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Number of Q-toothpicks added at n-th stage in the Q-toothpick structure of A267698.

We define a "Q-toothpick" to be a quarter-circle. The length of a Q-toothpick is equal to Pi/2 = 1.570796...

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

- Each new Q-toothpick must lie on the square grid (or circular grid) such that the Q-toothpick endpoints coincide with two opposite vertices of a unit square.

- Each exposed endpoint of the Q-toothpicks of the old generation must be touched by the endpoints of two q-toothpicks of new generation without creating a corner or vertex between these three arcs such that the couple of new Q-toothpicks should look like a "gullwing".

Note that in the Q-toothpick structure sometimes there is also an internal growth of the Q-toothpicks.

The sequence gives the number of Q-toothpicks in the structure after n stages. A187211 (the first differences) gives the number of Q-toothpicks added at n-th stage.

Note that the structure of the Q-toothpick cellular automaton contains distinct types of geometrical figures, for example: circles, diamonds, hearts, heads or flower vases (which appears only on the main diagonal) and also an infinity family of objects (blobs) where every object is a closed region which contains 2^k virtual circles with radius 1 and 2^k-1 virtual diamonds, for example: a 2 X 2 object is a closed region which contains exactly four virtual circles and three virtual diamonds, a 2 X 4 object is a closed region which contains exactly 8 virtual circles and 7 virtual diamonds, etc. Note that a "heart" can be considered a 1 X 2 object which contains two virtual circles and a virtual diamond. What is the better name for these figures? Note that there is a correspondence between this last family of objects and the squares and rectangles of the hidden crosses in the toothpick structure of A139250. For more information about the connection with the toothpick sequence see A139250, A160164 and A187220.

It appears that the number of hearts present in the n-th generation equals the number of rectangles of area = 2 present in the (n-2)nd generation of the toothpick structure of A139250, assuming the toothpicks have length 2, if n >= 3 (see also A188346 and A211008). - Omar E. Pol, Sep 30 2012

From Omar E. Pol, Jan 23 2016: (Start)

Consider the initial Q-toothpick with the virtual center at (0,0) and its endpoints at (0,1) and (1,0).

If n is a power of 2 plus 2 and n >> 1 then the structure of this C.A. essentially looks like a square which contains four parts (or sectors) as follows:

1) NW quadrant, but whose origin is at (-1,1). In this quadrant the number of Q-toothpicks after n generations equals the number of toothpicks in the toothpick structure of A139250 after n-2 generations, if n >= 2. Note that here the toothpick sequence A139250 is represented with Q-toothpicks arranged in an asymmetric structure.

2) SE quadrant, but whose origin is at (1,-1). This quadrant is a reflected copy of the NW quadrant, hence the number of Q-toothpicks after n generations equals A139250(n-2), n >= 2, the same as in the NW quadrant.

3) SW quadrant, but with the origin in the first quadrant at (1,1). In this quadrant the number of Q-toothpicks after n generations is 1 + A267694(n-1), n >= 1.

4) NE quasi-quadrant. In this sector the number of Q-toothpicks after n-generations is A267698(n-2) - 2, if n >= 6. (End)

After the first few generations the behavior is similar to the Gullwing cellular automaton of A187220, but the growth is faster than A187220 and thus it's much faster than A139250. For an animation see Applegate's The movie version in the Links section. - Omar E. Pol, Sep 13 2016

An H-toothpick sequence is a toothpick sequence on a square grid that resembles a partial honeycomb of hexagons.

The structure has two types of elements: the classic toothpicks with length 1 and the "D-toothpicks" with length sqrt(2).

Classic toothpicks are placed in the vertical direction and D-toothpicks are placed in a diagonal direction.

Each hexagon has area = 4.

The network looks like an elongated hexagonal lattice placed on the square grid so that all nodes of the hexagonal net coincide with some of the grid points of the square grid. Each node in the hexagonal network is represented with coordinates x,y.

The sequence gives the number of toothpicks and D-toothpicks after n steps. A182839 (first differences) gives the number added at the n-th stage.

[It appears that for this sequence a classic toothpick is a line segment of length 1 that is parallel to the y-axis. A D-toothpick is a line segment of length sqrt(2) with slope +-1. D stands for diagonal. It also appears that classic toothpicks are not placed on the y-axis. - N. J. A. Sloane, Feb 06 2023]

From Omar E. Pol, Feb 17 2023: (Start)

This cellular automaton appears to be a version on the square grid of the first quadrant of the structure of A182840.

The rules are as follows:

- The elements (toothpicks and D-toothpicks) are connected at their ends.

- At each free end of the elements of the old generation two elements of the new generation must be connected.

- The toothpicks of length 1 must always be placed vertically, i.e. parallel to the Y-axis.

- The angle between a toothpick of length 1 and a D-toothpick of length sqrt(2) that share the same node must be 135 degrees, therefore the angle between two D-toothpicks that share the same node is 90 degrees.

As a result of these rules we can see that in the odd-indexed rows of the structure are placed only the toothpicks of length 1 and in the even-indexed rows of the structure are placed the D-toothpicks of length sqrt(2).

Apart from the trapezoids, pentagons and heptagons that are adjacent to the axes of the first quadrant it appears that there are only three types of polygons:

- Regular hexagons of area 4.

- Concave decagons (or concave 10-gons) of area 8.

- Concave dodecagons (or concave 12-gons) of area 12.

There are infinitely many of these polygons.

The structure shows a fractal-like behavior as we can see in other members of the family of toothpick cellular automata.

The structure has internal growth as some members of the mentioned family. (End)

a(n) is the total number of Q-toothpicks in the structure after n-th stage.

A267695 (the first differences) gives the number of Q-toothpicks added at n-th stage.

a(n) is also the total number of Q-toothpicks minus 1 after n+2 stages in the SW quadrant of the Q-toothpick structure of A187210, but with origing at (1,1) instead (0,0), and assuming that the initial Q-toothpick is in the first quadrant with origin at (0,0), if n>=1 . For more information see the comments and the formula of A187210.

For the construction of this sequence we use the same the rules of A187210 (the Q-toothpick sequence) except that for the even-indexed generations here we use Q-toothpicks of radius 2, not 1.

The result is that the structure looks like an arrangement of ovals.

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

For n >= 1, a(n) is the ratio between the total length of the lines of the structure after n-th stages and the length of a single Q-toothpick of radius 1.

A187210(n) gives the total number of Q-toothpicks in the structure after n-th stages.

A187211(n) gives the number of Q-toothpicks added at n-th stage.

Note that since the radius of the Q-toothpicks can be two distincts numbers so we can write an infinite number of sequences from cellular automata of this kind.