The zero-multiplicity of third-order linear recurrences associated to the Tribonacci sequence

In this paper, we consider third-order linear recurrences { u n } n ≥ 0 satisfying the recurrence relation u n + 3 = u n + 2 + u n + 1 + u n for all n ≥ 0 and investigate the multiplicity of its zeros. We prove that { u n } n ≥ 0 has zero-multiplicity at most 2, except for nonzero multiples of shifts of the Tribonacci sequence which has zero-multiplicity 4 when the indices are extended to all the integers.