OFFSET
1,2
COMMENTS
The structure and the behavior of this cellular automaton reveals that some cellular automata have recurrent periods that can be represented by irregular triangles of first differences whose row lengths are the terms of A011782 multiplied by k (instead of powers of 2), where k is the length of their "word". In this case the word must be "abc", therefore k = 3. In the case of the cellular automaton with normal toothpicks (A139250) the word must be "ab" and k = 2.
The associated sound to the animation of this cellular automaton could be [tick, tock, tack], [tic, tock, tack], and so on.
For more information about the "word" of a cellular automaton see A296612.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..6144
Rémy Sigrist, Illustration of the construction at generation 3*256
Rémy Sigrist, PARI program
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
EXAMPLE
The structure of this irregular triangle is as shown below:
a, b, c;
a, b, c;
a, b, c, a, b, c;
a, b, c, a, b, c, a, b, c, a, b, c;
a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c, a, b, c;
...
Every column is associated successively to one of the axes of the triangular grid.
Every row represents a geometric period of the cellular automaton.
So, written as an irregular triangle in which the row lengths are the terms of A011782 multiplied by 3, the sequence begins:
1, 2, 4;
6, 6, 6;
6,10,16,20,16,10;
6,10,16,24,28,32,28,32,40,50,40,22;
8,10,16,24,28,32,32,40,56,74,76,64,42,36,40,62,76,90,80,88,102,122,96,50;
14,10,16,24,28,32,32,40,56,74,76,64,...
...
PROG
(PARI) See Links section.
CROSSREFS
First differences of A296510.
KEYWORD
AUTHOR
Omar E. Pol, Dec 14 2017
EXTENSIONS
More terms from Rémy Sigrist, Jul 22 2022
STATUS
approved