  Let $ \mathbb{Q}\mathcal{E}_{\mathbb{Z}} $ be the set of power sums whose
characteristic roots belong to $ \mathbb{Z} $ and whose coefficients belong to
$ \mathbb{Q} $, i.e. $ G : \mathbb{N} \rightarrow \mathbb{Q} $ satisfies
\begin{equation*} G(n) = G_n = b_1 c_1^n + \cdots + b_h c_h^n \end{equation*}
with $ c_1,\ldots,c_h \in \mathbb{Z} $ and $ b_1,\ldots,b_h \in \mathbb{Q} $.
Furthermore, let $ f \in \mathbb{Q}[x,y] $ be absolutely irreducible and $
\alpha : \mathbb{N} \rightarrow \overline{\mathbb{Q}} $ be a solution $ y $ of
$ f(G_n,y) = 0 $, i.e. $ f(G_n,\alpha(n)) = 0 $ identically in $ n $. Then we
will prove under suitable assumptions a lower bound, valid for all but finitely
many positive integers $ n $, for the approximation error if $ \alpha(n) $ is
approximated by rational numbers with bounded denominator. After that we will
also consider the case that $ \alpha $ is a solution of \begin{equation*}
f(G_n^{(0)}, \ldots, G_n^{(d)},y) = 0, \end{equation*} i.e. defined by using
more than one power sum and a polynomial $ f $ satisfying some suitable
conditions. This extends results of Bugeaud, Corvaja, Luca, Scremin and
Zannier.
