Analyzing discrete time signals
Since the function is periodic, it is also defined for negative arguments, e.g. \$x[-1]=x[7]\$ or \$x[-2]=x[6]\$, etc. That said, the notion of being odd makes sense.

Note that also the \$a_k\$ are periodic (in partiular \$a_9=a_1\$), and you get the following constraints:


being odd implies \$a_k\$ are imaginary
being real implies \$a_k=a^*_{-k}=a^*_{N-k}\$


Btw. for real DFTs, one normally takes advantage of the latter, see e.g. the FFTW docs: http://www.fftw.org/fftw3_doc/One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data

[Edited after comment below, thanks!]