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An old theorem of Pólya and Carlson [2] states that, if the power series 
has rational integer coefficients, positive radius of convergence, and can be continued analytically to a region that contains points outside the closed unit disc, then the function that the power series represents is rational. This result has been extended in a number of ways (cf. e.g. Petersson [4]). The present note gives a new extension based on a recent theorem of Güting [3]. My thanks to Professors Henry Helson and Raphael Robinson for introducing me to this subject.
We consider infinite sequences 
of positive integers having exponential growth: 
and becoming ultimately periodic modulo each member of a rather sparse infinite set of integers. If sufficient, natural conditions are placed on the growth and periodicities of , we find that a is an algebraic integer having all its algebraic conjugates within or on the unit circle, and fn has a special representation involving an. The result is a kind of dual to the theorem of Pisot (cf. Salem [2], p. 4, Theorem A).
