On Dold condition and fail factor of linear recurrent sequences

A sequence $\mathbf{A}$ is said to be realizable if satisfies so called sign and Dold conditions. We will say that a sequence almost satisfies the Dold condition if there exists a constant $c\in\mathbb{N}_+$ such that $(cA_n)_{n\in\mathbb{N}_+}$ satisfies the Dold condition. In this paper we give characterisation of sequences defined by linear recursion of any order that almost satisfy the Dold condition. We also give an upper bound on the value of $c$.