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.