Generating functions [for nested patterns]

A convenient algebraic way to describe a sequence of numbersa[n] is to give a generating functionSum[a[n] xⁿ, {n, 0,∞}].1/(1 - x) thus corresponds to the constant sequence and1/(1 - x - x²) to the Fibonacci sequence (see page890). A 2D array can be described bySum[a[t, n] xⁿ y^t, {n, -∞,∞}, {t, -∞,∞}]. The array for rule 60 is then1/(1- (1 + x) y), for rule 901/(1 - (1/x + x) y), for rule 1501/(1 - (1/x + 1 + x) y) and for second-order reversible rule 150 (see page439)1/(1 - (1/x + 1 + x) y - y²). Any rational function is the generating function for some additive cellular automaton.