  We study the $c$-vectors, $g$-vectors, and $F$-polynomials for generalized
cluster algebras satisfying a normalization condition and a power condition
recovering classical recursions and separation of additions formulas. We
establish a relationship between the $c$-vectors, $g$-vectors, and
$F$-polynomials of such a generalized cluster algebra and its (left- and
right-) companion cluster algebras. Our main result states that the cluster
variables and coefficients of the (left- and right-) companion cluster algebras
can be recovered via a specialization of the $F$-polynomials.
