Multidimensional continued fractions generalize classical continued fractions
with the aim of providing periodic representations of algebraic irrationalities
by means of integer sequences. However, there does not exist any algorithm that
provides a periodic multidimensional continued fraction when algebraic
irrationalities are given as inputs. In this paper, we provide a
characterization for periodicity of Jacobi--Perron algorithm by means of linear
recurrence sequences. In particular, we prove that partial quotients of a
multidimensional continued fraction are periodic if and only if numerators and
denominators of convergents are linear recurrence sequences, generalizing
similar results that hold for classical continued fractions
