  We consider a family of nonlinear recurrences with the Laurent property.
Although these recurrences are not generated by mutations in a cluster algebra,
they fit within the broader framework of Laurent phenomenon algebras, as
introduced recently by Lam and Pylyavskyy. Furthermore, each member of this
family is shown to be linearisable in two different ways, in the sense that its
iterates satisfy both a linear relation with constant coefficients and a linear
relation with periodic coefficients. Associated monodromy matrices and first
integrals are constructed, and the connection with the dressing chain for
Schrodinger operators is also explained.
