In mathematics, a locally catenative sequence is a sequence of words in which each word can be constructed as the concatenation of previous words in the sequence. [1]
Mathematics includes the study of such topics as quantity, structure, space, and change.
Formally, an infinite sequence of words w(n) is locally catenative if, for some positive integers k and i1,...ik:
Some authors use a slightly different definition in which encodings of previous words are allowed in the concatenation. [2]
The sequence of Fibonacci words S(n) is locally catenative because
A Fibonacci word is a specific sequence of binary digits. The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition.
The sequence of Thue–Morse words T(n) is not locally catenative by the first definition. However, it is locally catenative by the second definition because
In mathematics, the Thue–Morse sequence, or Prouhet–Thue–Morse sequence, is the binary sequence obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins:
where the encoding μ replaces 0 with 1 and 1 with 0.
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