OFFSET
0,2
COMMENTS
An Elementary Triangular Automaton (ETA) is a cellular automaton in the triangular grid where cells hold binary states and rules are local to the first neighborhood. There are 256 possible ETA rules.
Rule 210 (11010010 in binary): "cells with one 1 neighbor change state"
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|state of the cell |1|1|1|1|0|0|0|0|
|sum of the neighbors' states |3|2|1|0|3|2|1|0|
|cell's next state |1|1|0|1|0|0|1|0|
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This is the sequence of numbers formed in binary by the states that start at the initial 1 cell and follow a direction parallel to any of its sides.
The center of the grid stabilizes at n=5592 in an oscillator of period 14. Therefore, from this point on, the first binary digits of a(n) will be the same as those of a(n+14).
At some point in the sequence, the binary digits of a(n) repeat almost exactly every 4096 digits, starting from the 16622nd digit. See illustration of the binary digits of a(65534). I conjecture that exact periodicity is to be found later in the sequence, with a period that is a multiple of 12288.
LINKS
Paul Cousin, Table of n, a(n) for n = 0..2048
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Sequence plot for n = 0..8192 (binary digits, left-to-right)
Paul Cousin, Binary digits of a(65534) (left-to-right, top-to-bottom)
Paul Cousin, Triangular Automata
Paul Cousin, Rule 210
Paul Cousin, Triangular Automata Integer Sequences
Paul Cousin, Triangular Automata: The 256 Elementary Cellular Automata of the Two-Dimensional Plane, Complex Systems, 33(3), 2024, pp. 253-276.
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Cousin, May 21 2025
STATUS
approved