## A chaotic cousin of Conway’s recursive sequence.(English)Zbl 0959.05007

Summary: 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 $$Q(n)= Q(n- Q(n- 1))+ Q(n- Q(n- 2))$$, $$Q(1)= Q(2)= 1$$; see Douglas R. Hofstadter [Gödel, Escher, Bach: an eternal golden braid (1979; reprint 1981; Zbl 0457.03001)]. 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.

### MSC:

 05A15 Exact enumeration problems, generating functions 11B83 Special sequences and polynomials 11B37 Recurrences

Zbl 0457.03001
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### References:

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