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.