Properties
For given k and r, a set is k-automatic if and only if it is kr-automatic. Otherwise, for h and k multiplicatively independent, then a set is both h-automatic and k-automatic if and only if it is 1-automatic, that is, ultimately periodic.
If u(n) is a k-automatic sequence then the sequences u(kn) and u(kn−1) are ultimately periodic. Conversely, if v(n) is ultimately periodic then the sequence u defined by u(kn) = v(n) and otherwise zero is k-automatic.
Let u(n) be a k-automatic sequence over the alphabet A. If f is a uniform morphism from A∗ to B∗ then the word f(u) is k-automatic sequence over the alphabet B.
Let u(n) be a sequence over the alphabet A and suppose that there is an injective function j from A to the finite field Fq. The associated formal power series is
The sequence u is q-automatic if and only if the power series fu is algebraic over the rational function field Fq(z).
Read more about this topic: Automatic Sequence
Famous quotes containing the word properties:
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—John Locke (1632–1704)
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—Ralph Waldo Emerson (1803–1882)