View source for Fibonacci word

An infinite sequence of symbols over a binary alphabet defined by a recursion related to that of the [[Fibonacci numbers]].

Let $A$ be the alphabet $\{a,b\}$ and define a sequence of finite words $f_n$, $n \ge 1$ by $f_1 = a$, $f_2 = ab$ and $f_{n+1} = f_n \cdot f_{n-1}$ where $\cdot$ denotes concatenation.  The Fibonacci word $\mathcal{F}$ is defined as
$$
\mathcal{F} = f_1 \cdot f_2 \cdot \cdots \cdot f_n \cdot \cdots
$$
and, since $f_n$ is an initial factor of $f_{n+1}$ we may also obtain $\mathcal{F}$ as the limit of the $f_n$.  We have
$$
\mathcal{F} = a ab aab abaab aababaab abaabaababaab  \ldots \ .
$$

The Fibonacci word has complexity $c_{\mathcal{F}}(n) = n+1$ and hence is a [[Sturmian sequence]].  It is a [[morphic word]], a fixed point of the endomorphism $a \mapsto ab$, $b \mapsto a$.

====References====
* Allouche, Jean-Paul; Shallit, Jeffrey ''Automatic Sequences: Theory, Applications, Generalizations,'' Cambridge University Press (2003) ISBN 978-0-521-82332-6 {{ZBL|1086.11015}}
* Lothaire, M. ''Combinatorics on Words'' (2nd ed.) Encyclopedia of Mathematics and Its Applications '''17''''  Cambridge University Press (1997) ISBN 0-521-59924-5 {{ZBL|0874.20040}}
* Lothaire, M. ''Algebraic Combinatorics on Words'' Encyclopedia of Mathematics and Its Applications '''90''', Cambridge University Press (2011 [2002]) ISBN 978-0-521-18071-9 {{ZBL|1221.68183}}
* Pytheas Fogg, N. (ed.) ''Substitutions in dynamics, arithmetics and combinatorics'' Lecture Notes in Mathematics '''1794''' Springer (2002) ISBN 978-3-540-44141-0 {{ZBL|1014.11015}}

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