On the zero–multiplicity of a fifth-order linear recurrence

We consider a family of linear recurrence sequences [Formula: see text] of order [Formula: see text] whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. In this paper, we study the zero–multiplicity on [Formula: see text] when the indices are extended to all integers. In particular, we give a upper bound (dependent on [Formula: see text]) for the largest positive integer [Formula: see text] such that [Formula: see text] and show that [Formula: see text] has zero–multiplicity unitary when the indices are extended to all the integers.