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Abstract

A ternary linear recurrence $(u_n)_{n\ge 0}$ is of Berstel type if it satisfies the recurrence relation $u_{n+3}=2u_{n+2}-4u_{n+1}+4u_{n}$ for all $n\ge 0$. In this paper, we investigate the zero-multiplicity of such sequences. We prove that, except for nonzero multiples of shifts of the Berstel sequence with initial values $0,0,1$, which has zero-multiplicity 6, and nonzero multiples of shifts of the sequence with initial values $0,1,4$, which has zero-multiplicity 3, all other sequences have zero multiplicity at most 2.