User:DroneBetter/OEIS elementary index
This is a table intended to be more convenient to use than the OEIS wiki's Index to Elementary Cellular Automata for sequences describing rules' behaviour beginning from states consisting of single on-cells in infinite planes.
Rules are grouped by equivalence class instead of coinciding single-cell evolution, and those on adjacent rows in the same class that have the same sequence for any of the columns have them merged into 2-row sections.
It doesn't contain links to the sequences describing the evolution of the central column, since they are trivial and unuseful for all but the chaotic rules.
Key:
- invariances
- N: anisotropic and not self-complementary. (4 members of each class)
- S: isotropic but not self-complementary. (2 members)
- I: self-complementary but not isotropic. (4 members)
- O: Both self-inverse and symmetrical. (2 or 4 members)
- A: Equivalent to its black/white reversal when reflected. (2 members)
- rule colour
- nothing: background remains invariant
- red: background becomes off
- green: background becomes on
- blue: background strobes
- characteristic
- p: all sequences of cell states on integer points along straight-line paths through evolution sequence are linear-recurrent[1]
- 1: all finite states eventually become entirely off
- l: all finite states eventually become comprised of finitely many noninteracting lines
- 2: lines must be vertical columns
- f: fractal
- s: Sierpinski
- c: chaotic
- note that all OEIS sequences are linear-recurrent (the arrays multivariate) for all but the fractal and chaotic rules
- equivalence operators
- r: black/white reversal
- f: left/right reflection
- s: strobing duality
- c: chequerboard duality (XOR each transition's input with 0b101)
within each invariance set, equivalence classes are sorted by their minimal representative, with exceptions to make leading members even where possible and make the equivalence operators more regular
alike the OEIS, every sequence with an asterisk requires an offset be imposed to match the actual sequence
notes
- ↑ if one were to create a category requiring only that each column is periodic, rule 60/102 would satisfy this despite forming a right-aligned Sierpinski triangle
- ↑ the limit of the evolution of rule 106/120 from a domino or 169/225 from a single state is self-similar, under the condition that it be compressed vertically when shrunk other initial states appear to usually result in chaos
- ↑ takes 2854 generations before on-cell counts' first differences begin periodic oscillations; every column and leftward diagonal is a linear-recurrent sequence of 0s and 1s
- ↑ from a single cell, 73 behaves aperiodically, 109 forms an agar; from a random infinite state, all cells have an indestructible isolated domino on either side of them with probability 1, between which they may only behave periodically
see also
- One-dimensional cellular automaton/Wolfram rule (also a table alike this)