Complete padovan sequences in finite fields

Given a prime p ≥, and given 1 < κ < p-1, we call a sequence (a_n)_n in F_p a Φκ-sequence if it is periodic with period p-1, and if it satisfies the linear recurrence a _n + a_n+1 - a_n+κ with a₀ = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a₀,a₁, ⋯, a_p-2} = {1, ⋯,p-1}. For instance, every primitive root b mod p generates a complete Φ_κ-sequence a_n = bⁿ for some (unique) κ. A natural question is whether every complete Φ_κ- sequence is necessarily defined by a primitive root. For κ = 2 the answer is known to be positive. In this paper we reexamine that case and investigate the case κ = 3 together with the associated cases κ = p - 2 and κ = p - 3.