A sequence \(g:\mathbb{N}\to\mathbb{C}\) is called almost-periodic if it belongs to the completion \({\mathcal A}^1\) of the \(\mathbb{C}\)-linear space spanned by the sequences \(e_\vartheta\) with \(\vartheta\in\mathbb{R}/ \mathbb{Z}\), where \(e_\vartheta(n) =e^{2\pi i\vartheta n}\) for \(n\in\mathbb{N}\), under the semi-norm \[ |g|_1= \limsup_{x\to\infty} {1\over x} \sum_{n\leq x} \bigl|g(n) \bigr|. \] Every \(g\in {\mathcal A}^1\) has a mean-value \[ M(g)= \lim_{x\to\infty} {1\over x} \sum_{n\leq x} g(n). \] A sequence \(g:\mathbb{N} \to\mathbb{C}\) is called recurrent if it satisfies a linear recurrence equation of the form \[ g(n+k) +a_{k-1} g(n+k-1)+ \cdots+ a_0g(n)=0 \quad (n\in \mathbb{N},\;n>n_0) \] with coefficients \(a_{k-1}, \dots, a_0\in\mathbb{C}\), \(a_0\neq 0\), and with some numbers \(k\), \(n_0\in\mathbb{N} \cup\{0\}\). Let \({\mathcal R}\) denote the space of recurrent sequences. It is shown that a sequence \(g\in{\mathcal A}^1\) cannot belong to \({\mathcal R}\) if \(M (ge_\vartheta)\neq 0\) for infinitely many \(\vartheta\in\mathbb{R}/ \mathbb{Z}\), which extends a recent result of \textit{J. Spilker} [ibid. 66, 292-298 (1996; Zbl 0855.11003)]. The proof is based on Kronecker's rationality test.