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But it turns out that with slightly more complicated rules it is possible to get much more complicated behavior. The key idea is to consider rules which look at numbers that are not just a fixed distance back in the sequence. And what this means is that instead of depending only on quantities like f[n-1] and f[n-2], the rule for f[n] can also for example depend on a quantity like f[n - f[n-1]].

There is some subtlety here because in the abstract nothing guarantees that n-f[n-1] will necessarily be a positive number. And if it is not, then results obtained by applying the rule can involve meaningless quantities such as f[0], f[-1] and f[-2].

Captions on this page:

Examples of sequences generated by rules that do not depend only on elements a fixed distance back. Most such rules eventually end up involving meaningless quantities such as f[0] and f[-1], but the particular rules shown here all avoid this problem.

From Stephen Wolfram: A New Kind of Science [citation]