OFFSET
0,3
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.
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.
LINKS
EXAMPLE
Illustration of initial terms:
.
/__\
_\ /_ _\ /_
/__\ /__\ /\/__\/\
/\ _\/\/_ _\/\/_ _\/\/_ /__\/\/__\
/\ /\ _\/\/__\/\/_ _\/\/__\/\/_
/\ /\
.
n: 1 2 3 4 5
a(n): 1 3 7 13 21
.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jean Hoffmann and Omar E. Pol, Jul 20 2022
STATUS
approved