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Recurrence relations
The rules for the sequences given here all have the form of linear recurrence relations.
[Repetition in] systems based on numbers
An iterated map of the kind discussed on page 150 with rule x Mod[a x, 1] (with rational a ) will yield repetitive behavior when its initial condition is a rational number.
A general feature of cellular automaton rules is that they are fundamentally local.
For n = 2 , the largest number of such operations is 6, achieved by Nor ; for n = 3 , it is 14, achieved by Xor (rule 150); for n = 4 , it is 27, achieved by rule 5737, which is Not[Xor[##]] & except when all inputs are True .
[Cellular automaton] rules based on algebraic systems
If the values of cells are taken to be elements of some finite algebraic system, then one can set up a cellular automaton with rule
a[t_, i_] := f[a[t - 1, i - 1], a[t - 1, i]]
where f is the analog of multiplication for the system (see also page 1094 ).
Forms of living systems
This book has shown that even with underlying rules of some fixed type a vast range of different forms can often be produced.
(m) The pattern can be generated by a 2D substitution system with rule {1 -> {{0, 0}, {0, 1}}, 0 -> {{1, 1}, {1, 0}}} (see page 583 ).
(a), (b) and (c) correspond respectively to the rules on pages 73 , 74 and 75 .
As discussed on page 903 this sequence can be generated by applying substitution rules derived from the continued fraction form of h .
This can be achieved by having more complicated underlying rules.