Consider constructing a vector $v=\{a_1,a_2,\ldots,a_n\}$ consisting of nonnegative integers such that $a_1=1$ and, if $a_j$'s are nonzero , then $a_j\equiv a_{n-j+2}+j-1 \pmod m\ \forall j\neq1$, where $m$ is the number of nonzero entries; with the additional constraint that all nonzero $a_i$'s are distinct.

Is it always possible to construct such a vector? I think this should be possible if $m$ is odd. For example, $\{1,4,2,5,3\}$ and $\{1,4,2,0,0,5,3\}$ are such vectors. It is easy to construct if the first entries (within $\lfloor\frac{n}{2}\rfloor$ )of the vector are consecutive. But, in other cases, it is not clear as to how to proceed with the construction. Any hints? Thanks beforehand.