Definition:Fibonacci String

Definition
Consider the alphabet $\set {\text a, \text b}$.

For all $n \in \Z_{>0}$, let $S_n$ be the (finite) string formed as:


 * $S_n = \begin {cases} \text a & : n = 1 \\

\text b & : n = 2 \\ S_{n - 1} S_{n - 2} & : n > 2 \end{cases}$

where $S_{n - 1} S_{n - 2}$ denotes that $S_{n - 1}$ and $S_{n - 2}$ are concatenated.

The terms of the sequence $\sequence {S_n}$ are Fibonacci strings.

Also defined as
The specific alphabet used is immaterial.

Different sources use different alphabets; $\set {0, 1}$ is another common one.

Different starting values are also sometimes seen, for example:
 * $S_0 = 0, S_1 = 01$

uses $\set {\text a, \text b}$ and the starting values given, by preference and for internal consistency.

Also known as
Some sources refer to a Fibonacci string as a Fibonacci word.