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And indeed sequence (a) is certainly not random; in fact it is purely repetitive. And in general it is fairly easy to see that in any sequence that is purely repetitive there must beyond a certain length be many blocks whose frequencies are far from equal.

It turns out that the same is true for nested sequences. And in the picture above, sequences (b), (c) and (d) are all nested.

But what about the remaining sequences? Sequences (e) and (f) seem to yield frequencies that in every case correspond accurately to those obtained by averaging over all possible sequences. Sequences (g) and (h) yield results that are fairly similar, but exhibit some definite fluctuations.

So do these fluctuations represent evidence that sequences (g) and (h) are not in fact random? If one looks at the set of all possible sequences, one can fairly easily calculate the distribution of frequencies for any particular block. And from this distribution one can tell with