Answer
$2^{\frac{n}{2}}$ if n is even and $2^{\frac{n+1}{2}}$ if n is odd.
Work Step by Step
a palindrome of length n is completely determined by its first $\left \lceil{\frac{n}{2}}\right \rceil$ bits.
This is true because once these bits are specified, the remaining bits, read from right to left, must be identical
to the first $\left \lfloor{\frac{n}{2}}\right \rfloor$ bits, read from left to right. Furthermore, for each bit we have 2 options,
Therefore, by the product rule there are $\left \lceil{\frac{n}{2}}\right \rceil $ways to do so.
i.e., $2^{\frac{n}{2}}$ if n is even and $2^{\frac{n+1}{2}}$ if n is odd.
