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(The second example is analogous to the Fibonacci substitution system on page 83 .)
(c) (Fibonacci-related sequence) The sequence at step t can be obtained from a[t_] := Join[a[t - 1], a[t - 2]]; a[1] = {0}; a[2] = {0, 1} . This sequence has length Fibonacci[t + 1] (or approximately 1.618 t + 1 ) (see note below ). … The ratio of the number of white elements to black at step t is Fibonacci[t - 1]/Fibonacci[t - 2] , which approaches GoldenRatio for large t .
So long as f[n] grows less rapidly than 2 n (as when f = Fibonacci or f = Prime ), digits 0 and 1 will suffice, though the representation is not generally unique.
My experience has usually been that addition is easiest, followed by multiplication, powers, Fibonacci numbers, perfect numbers and then primes.
Rule (c) is related to the Fibonacci sequence.
In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.
The number of distinct sequences at step t in these three systems is respectively Ceiling[t/2] , t and Fibonacci[t+1] (which increases approximately like 1.618 t ).
Multiway systems based on numbers
One can consider for example the rule n {n + 1, 2 n} implemented by
NestList[Union[Flatten[{# + 1, 2 #}]] &, {0}, t]
In this case there are Fibonacci[t + 2] distinct numbers obtained at step t .
As an alternative to modelling individual organisms, one can also consider substitution systems which directly generate genealogical trees for populations of organisms, somewhat like Leonardo Fibonacci 's original model of a rabbit population.
Generating functions [for nested patterns]
A convenient algebraic way to describe a sequence of numbers a[n] is to give a generating function Sum[a[n] x n , {n, 0, ∞ }] . 1/(1 - x) thus corresponds to the constant sequence and 1/(1 - x - x 2 ) to the Fibonacci sequence (see page 890 ).