Let $q\ge 2$ be a positive integer and let $\left(a_j\right)$ , $\left(b_j\right)$ , and $\left(c_j\right)$ (with j a non-negative integer) be three given $\mathbb{C}$ -valued and q-periodic sequences. Let $A\left(q\right):=A_{q-1}\cdots A_0$ , where $A_j$ is as is given below. Assuming that the “monodromy matrix” $A\left(q\right)$ has at least one multiple eigenvalue, we prove that the linear scalar recurrence $x_{n+3}=a_n{x}_{n+2}+b_n{x}_{n+1}+c_n{x}_n,n\in {\mathbb{Z}}_{+}$ is Hyers-Ulam stable if and only if the spectrum of $A\left(q\right)$ does not intersect the unit circle $\mathsf{\Gamma }:=\{w\in \mathbb{C}:|w|=1\}$ . Connecting this result with a recently obtained one it follows that the above linear recurrence is Hyers-Ulam stable if and only if the spectrum of $A\left(q\right)$ does not intersect the unit circle.

Let $q\ge 2$ be a positive integer and let $\left(a_j\right),\left(b_j\right)$ and $\left(c_j\right)$ (with j nonnegative integer) be three given $\mathbb{C}$ -valued and q-periodic sequences. Let $A\left(q\right):=A_{q-1}\cdots A_0,$ where $A_j$ is defined below. Assume that the eigenvalues $x,y,z$ of the “monodromy matrix” $A\left(q\right)$ verify the condition $(x-y)(y-z)(z-x)\ne 0.$ We prove that the linear recurrence in $\mathbb{C}$ $x_{n+3}=a_n{x}_{n+2}+b_n{x}_{n+1}+c_n{x}_n,n\in {\mathbb{Z}}_{+}$ is Hyers–Ulam stable if and only if $\left(\right|x|-1)\left(\right|y|-1)\left(\right|z|-1)\ne 0,$ i.e., the spectrum of $A\left(q\right)$ does not intersect the unit circle $\mathsf{\Gamma }:=\{w\in \mathbb{C}:|w|=1\}.$