Unary languages recognized by two-way deterministic counter automata 2dca's (two-way deterministic one-counter automata) (Petersen, 1994) can recognize the following unary language:
\begin{equation} 
\mathtt{POWER} = \lbrace  0^{2^n} \mid n \geq 0 \rbrace.
\end{equation}

Is there any other nontrivial unary language recognized by 2dca's?

Remark that it is still unknown whether 2dca's can recognize $ \mathtt{SQUARE} = \lbrace  0^{n^2} \mid n \geq 0 \rbrace $?



DEFINITION: A 2dca is a two-way deterministic finite automaton with a counter. A 2dca can test whether the value of the counter is zero or not, and increment or decrement the value of the counter by 1 in each step.
This is only an idea that came to my mind while reading Marvin L. Minsky, "Recursive Unsolvability of Post's Problem of Tag and other Topics in Theory of
Turing Machines"; in particular the famous theorem Ia:


Theorem Ia: We can represent any partial recursive function $f(n)$
by a program operating on two integers $S_1$ and $S_2$
using instructions $I_j$ of the forms:
(i) ADD 1 to $S_j$, and go to $I_{j_1}$
(ii) SUBTRACT 1 from $S_j$, if $S_j \neq 0$ and go to $I_{j_1}$, otherwise go to $I_{j_2}$
That is, we can construct such a program that starts with $S_1 = 2^n$ and $S_2 = 0$ and
eventually stops with $S_1 = 2^{f(n)}$ and $S_2 = 0$


If you have a two way DFA with one counter over a (semi)infinite tape where the input is given in unary: $\$1^{2^n}000...$ then the DFA can:


read the unary input (and store it in the counter);
work on the $0^\infty$ part of the tape and use the distance from the $1$s as the second counter.


so it can simulate a Turing complete two counters machine.

Now, if you have a recursive function $f(n)$ that runs in time $T(n)$ on a standard Turing machine, a two way DFA with one counter that starts on the finite tape $\$1^m\$ \;$ (where $m = 2^n3^{T'(n)}$ and $T'(n) \gg T(n)$) can:


read the unary input (and store it in the counter);
return to leftmost symbol;
divide the counter by 3 until the counter contains $2^n$ in this way: go right looping from states $q_{z_0}, q_{z_1}, q_{z_2}$ and subtracting 1; if counter reaches 0 in state $q_{z_0}$ go to leftmost symbol adding +1 and continue the division loop, otherwise add 1 (if in state $q_{z_1}$) or 2 (if in state $q_{z_2}$) and go to leftmost symbol adding +3 (i.e. recover the previous value of the counter not divisible by 3) and proceed with step 4.;
at this point the counter contains $2^n$;
calculate $2^{f(n)}$ using the $T'(n)$ space available on the right as the second counter (the value of the second counter is the distance from the leftmost symbol $\$$).


So with the special input encoding described above that gives it enough space on the finite tape, a two-way DFA with one counter and unary alphabet can compute every recursive function.

If the approach is correct, it would be interesting to reason about how to choose $T'(n) \gg T(n)$ or when it is enough to pick a large odd $k \gg 2$ and encode the input as $1^m$, $m = 2^n k^n$