 The next step from a vocabulary consisting of only monadic predicates is a vocabulary with exactly one unary function. To describe some highly non-trivial results in this case we introduce the following concept, which has independent interest: The spectrum of a first or second-order sentence is the set of cardinalities of its finite models. Spectra were introduced by Scholz, see Durand, Jones, Makowsky, and More (2012) for a history. A spectrum is of course always recursive. In fact, a set of natural numbers is a spectrum of a first order sentence iff it is in NEXPTIME, that is, recognizable by a nondeterministic Turing machine in exponential time (Jones &amp; Selman 1974). In the case of sentences with just one unary function symbol there is a surprising characterization. Let us call a set S of natural numbers eventually periodic if there are natural numbers N and p with \(p &gt; 0\) such that for every n with \(n &gt; N\), we have \(n \in S\) iff \(n +p \in S\).