  A two-way deterministic finite state automaton with one counter (2D1CA) is a
fundamental computational model that has been examined in many different
aspects since sixties, but we know little about its power in the case of unary
languages. Up to our knowledge, the only known unary nonregular languages
recognized by 2D1CAs are those formed by strings having exponential length,
where the exponents form some trivial unary regular language. In this paper, we
present some non-trivial subsets of these languages. By using the input head as
a second counter, we present simulations of two-way deterministic finite
automata with linearly bounded counters and linear--space Turing machines. We
also show how a fixed-size quantum register can help to simplify some of these
languages. Finally, we compare unary 2D1CAs with two--counter machines and
provide some insights about the limits of their computational power.
