Fibonacci Length of an Efficiently Presented Metabelian p-Group

For every integer $ngeqslant 1$ and every odd prime $p$, anefficient presentation is given for the wreath product$mathbb{Z}_pwr mathbb{Z}_{p^n}$ which, is of nilpotency class$(p-1)n+1$. Moreover, the Fibonacci length is proved to be$8 imes 3^n$ when $p=3$, and in general, $k(p) imes p^n$, where$k(p)$ is the period of Fibonacci length modulo $p$ (the Wallnumber of $p$).