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

We consider a family of linear recurrence sequences T(k):= {T n(k)} n of order k whose first k terms are 0, 1,.., 1 and each term afterwards is the sum of the preceding k terms. In this paper, we study the zero-multiplicity on T(k) when the indices are extended to all integers. In particular, we give a upper bound (dependent on k) for the largest positive integer n such that T-n(k) = 0 and show that T(5) has zero-multiplicity unitary when the indices are extended to all the integers.