Notes

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] xn, {n, 0, }]. 1/(1 - x) thus corresponds to the constant sequence and 1/(1 - x - x2) to the Fibonacci sequence (see page 890). A 2D array can be described by Sum[a[t, n] xn yt, {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 - y2). Any rational function is the generating function for some additive cellular automaton.

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