A special case of rational θs for terminating θ-expansions

There have been quite a few generalizations of the usual continued fraction expansions over the last few years. One very special generalization deals with θ-continued fraction expansions or simply θ-expansions introduced by Bhattacharya and Goswami [A class of random continued fractions with singular equillibria, Perspectives in Statistical Science. eds A.K.Basu et al, Oxford University Press, 2000]. Chakraborty and Rao [θ-expansions and the generalized Gauss map, Probability, Statistics and their Applications: Papers in Honor of Rabi Bhattacharya. eds Athreya, K. et al, IMS Lect. Notes, Monogr. Ser. 41 (2003)] subsequently did elaborate studies on θ-expansions in their paper. They also obtained the unique invariant measure for the Markov process associated with the generalized Gauss transformation that generated θ-expansions for some special θs. In this work, we investigate an interesting question regarding the nature of θs for θ-expansion of 1/θ terminating at stage two, particularly with θ rational