A355310 Total number of V-toothpicks after n-th stage in a cellular automaton with V-toothpicks of 60 degrees (see Comments lines for precise definition).

0, 1, 3, 7, 13, 21, 27, 37, 51, 69, 79, 89, 103, 123, 141, 165, 201, 245, 267

Offset: 0

Author

Jean Hoffmann and Omar E. Pol, Jul 20 2022

Keywords

• nonn
• more

Comments

An idea from Jean Hoffmann.

In this cellular automaton a V-toothpick is formed by 2 toothpicks of length 1 that share a vertex and the angle between both toothpicks is 60 degrees.

On the infinite triangular grid we start with no V-toothpick, so a(0) = 0.

At stage 1 we place a V-toothpick upside down, so a(1) = 1.

At every stage the V-toothpicks of the new generation must be connected to the structure by touching with their middle vertex the free ends of the V-toothpicks of the previous generation following a special rule:

The new V-toothpicks must be placed between the imaginary straight line containing the two extreme ends of the V-toothpick of the previous generation and the imaginary straight line that contains the middle vertex of that V-toothpick and that it is parallel to the aforementioned straight line.

A355311(n) gives the number of V-toothpicks added to the structure at the n-th stage.

2*a(n) is the total number of toothpicks of length 1 in the structure after n-th stage.

This cellular automaton is a companion of the Y-toothpick cellular automaton of A160120 in the sense that both essentially grow as an equilateral triangle.

This cellular automaton is slightly less symmetrical than Y-toothpick cellular automaton because its structure has a "backbone" formed by concave hexagons from the center of the triangle to one of its vertices.

The behavior could be very close to A160120 and similar to A153006 (see the graph).

After 18 stages we can see in the structure the following polygons:

- Equilateral triangles of perimeter 3.

- Equilateral triangles of perimeter 6 that contain 4 triangular cells.

- Concave hexagons of perimeter 8 that contain 6 triangular cells.

- Concave dodecagons (or concave 12-gons) of perimeter 18 that contain 22 triangular cells.

Examples

			Illustration of initial terms:
.
                                                                  /__\
                                               _\  /_            _\  /_
                                 /__\           /__\            /\/__\/\
            /\      _\/\/_      _\/\/_         _\/\/_          /__\/\/__\
                               /\    /\     _\/\/__\/\/_      _\/\/__\/\/_
                                                             /\          /\
.
  n:         1         2           3              4                 5
  a(n):      1         3           7             13                21
.
		

Links

• Jean Hoffmann and Omar E. Pol, Illustration of a(18) = 267
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A355311 (first differences).

Cf. A139250, A153006, A160120, A161206, A161412, A161420, A299476, A299478, A327330, A327332.