Complete Padovan Sequences in Finite Fields

Given a prime p ≥ 5, and given 1 < κ < p − 1, we call a sequence (an)n in Fp a Φκ-sequence if it is periodic with period p− 1, and if it satisfies the linear recurrence an + an+1 = an+κ with a0 = 1. Such a sequence is said to be a complete Φκ-sequence if in addition {a0, a1, . . . , ap−2} = {1, . . . , p− 1}. For instance, every primitive root b mod p generates a complete Φκ-sequence an = b n for some (unique) κ. A natural question is whether every complete Φκ-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.