A CHAOTIC COUSIN OF CONWAY’S RECURSIVE SEQUENCE

KLAUS PINN

Abstract. I introduce the recurrence

D(n) = D(D(n−1)) + D(n−1 − D(n−2)),

D(1) = D(2) = 1, and study it by means of computer experiments. The definition of D(n) has some similarity to that of Conway’s sequence defined by a(n) = a(a(n−1)) + a(n − a(n−1)), a(1) = a(2) = 1. However, unlike the completely regular and predictable behaviour of a(n), the D-numbers exhibit chaotic patterns. In its statistical properties, the D-sequence shows striking similarities with Hofstadter’s Q(n)-sequence, given by Q(n) = Q(n − Q(n−1)) + Q(n − Q(n−2)), Q(1) = Q(2) = 1. Compared to the Hofstadter sequence, D shows higher structural order. It is organized in well-defined “generations”, separated by smooth and predictable regions. The article is complemented by a study of two further recurrence relations with definitions similar to those of the Q-numbers. There is some evidence that the different sequences studied share a universality class.