Additive cellular automata and algebraic series

Abstract

A cellular automation is an array of regularly interconnected identical cells. We study here the special case of automata where each cell depends in additive manner on its neighbours. The successive states of a given cell form a sequence whose generating series is proved to be always an algebraic series. We also exemplify the realization of a given algebraic series by means of an automation. As a by-product we obtain a relation between additive cellular automata and certain “automatic sequences” like the paper-folding sequence.