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]
Formally, an infinite sequence of words w(n) is locally catenative if, for some positive integers k and i1,...ik:
w(n)=w(n-i1)w(n-i2)\ldotsw(n-ik)forn\gemax\{i1,\ldots,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
S(n)=S(n-1)S(n-2)forn\ge2.
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
T(n)=T(n-1)\mu(T(n-1))forn\ge1,