The purpose of this paper is to show connections between iterated rational transductions and linear space computations. Hence, we study the smallest family of transductions containing length-preserving rational transductions and closed under union, composition and iteration. We give several characterizations of this class using restricted classes of length-preserving rational transductions, by showing the connections with ``context-sensitive transductions'' and transductions associated with recognizable picture languages.

We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.