Complete Padovan sequences in finite fields

Given a prime $p\ge 5$, and given $1<\kappa <p-1$, we call a sequence $(a_n{)}_n$ in ${\mathbb{F}}_p$ a ${\mathrm{\Phi }}_{\kappa }$-sequence if it is periodic with period $p-1$, and if it satisfies the linear recurrence $a_n+a_{n+1}=a_{n+\kappa }$ with $a_0=1$. Such a sequence is said to be a complete ${\mathrm{\Phi }}_{\kappa }$-sequence if in addition $\{a_0,a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_{p-2}\}=\{1,\mathrm{.}\mathrm{.}\mathrm{.},p-1\}$. For instance, every primitive root $b$ mod $p$ generates a complete ${\mathrm{\Phi }}_{\kappa }$-sequence $a_n=b^n$ for some (unique) $\kappa$. A natural question is whether every complete ${\mathrm{\Phi }}_{\kappa }$-sequence is necessarily defined by a primitive root. For $\kappa =2$ the answer is known to be positive. In this paper we reexamine that case and investigate the case $\kappa =3$ together with the associated cases $\kappa =p-2$ and $\kappa =p-3$.