  Let $F$ be the Fibonacci matrix $ \bigl[\begin{smallmatrix} 1 & 1 1 & 0 \\
\end{smallmatrix}\bigr] $. The Fibonacci Dyck shift is a subshsystem of the
Dyck shift $D_2$ constrained by the matrix $F$. Let ${{\frak L}^{Ch(D_F)}}$ be
a $\lambda$-graph system presenting the subshift $D_F$, that is called the
Cantor horizon $\lambda$-graph system for $D_F$. We will study the
$C^*$-algebra ${\cal O}_{{\frak L}^{Ch(D_F)}}$ associated with $ {{\frak
L}^{Ch(D_F)}} $. It is simple purely infinite and generated by four partial
isometries with some operator relations. We will compute the K-theory of the
$C^*$-algebra. As a result, the $C^*$-algebra is simple purely infinite and not
semiprojective. Hence it is not stably isomorphic to any Cuntz-Krieger algebra.
