Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas

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). A 2D array can be described by Sum[a[t, n] x^n y^t, {n, -∞, ∞}, {t, -∞, ∞}]. The array for rule 60 is then 1/(1-(1+x) y), for rule 90 1/(1-(1/x+x) y), for rule 150 1/(1-(1/x+1+x)y) and for second-order reversible rule 150 (see page 439) 1/(1-(1/x+1+x)y-y^2). Any rational function is the generating function for some additive cellular automaton.

From Stephen Wolfram: A New Kind of Science [citation]