cp's OEIS Frontend

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)

See the comments in A161644.

It appears that a(n) is also the number of V-toothpicks or Y-toothpicks added at the n-th stage in a toothpick structure on hexagonal net, starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >=2 (see A161206, A160120, A182633). - Omar E. Pol, Dec 07 2010

First differences of A182840.

The sequence gives the number of toothpicks after n stages. A182635 (the first differences) gives the number added at the n-th stage.

The 120-degree wedge defines a conic region which toothpicks (except one end point of the initial toothpick) are not allowed to cross or touch. The wings of the wedge point +-60 degrees away from the pointing direction of the initial toothpick.

Toothpicks are connected by their endpoints, the same as the toothpicks of A182632.

First differs from A139250 at a(11).

First differences of A182634.

First differs from A139251 at a(11).

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)

Essentially the first differences of A222180.

Also number of P-toothpicks added at n-th stage to the P-toothpick structure of A222180.

Essentially the first differences of A233970.

First differs from A170905 at a(24).

First differs from both A233765 and A233781 at a(25).

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.