squarefree sequence
A squarefree sequence is a sequence which has no adjacent repeating subsequences^{} of any length.
The name “squarefree” comes from notation: Let $\{s\}$ be a sequence. Then $\{s,s\}$ is also a sequence, which we write “compactly” as $\{{s}^{2}\}$. In the rest of this entry we use a compact notation, lacking commas or braces. This notation is commonly used when dealing with sequences in the capacity of strings. Hence we can write $\{s,s\}=ss={s}^{2}$.
Some examples:

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$xabcabcx=x{(abc)}^{2}x$, not a squarefree sequence.

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$abcdabc$ cannot have any subsequence written in square notation, hence it is a squarefree sequence.

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$ababab={(ab)}^{3}=ab{(ab)}^{2}$, not a squarefree sequence.
Note that, while notationally similar to the numbertheoretic sense of “squarefree,” the two concepts are distinct. For example, for integers $a$ and $b$ the product $aba={a}^{2}b$, a square. But as a sequence, $aba=\{a,b,a\}$; clearly lacking any commutativity that might allow us to shift elements. Hence, the sequence $aba$ is squarefree.
Title  squarefree sequence 

Canonical name  SquarefreeSequence 
Date of creation  20130322 11:55:36 
Last modified on  20130322 11:55:36 
Owner  akrowne (2) 
Last modified by  akrowne (2) 
Numerical id  12 
Author  akrowne (2) 
Entry type  Definition 
Classification  msc 11B83 
Synonym  square free sequence 