Fourier series period?
The period is indeed \$N\$: the sample at index \$n=N\$ corresponds to the sample at \$n=0\$. So if you continued the sequence to the left (i.e. towards negative indices), there would be three samples with value \$1\$.

For finding the coefficients you need to sum over one period, i.e. you have to sum from \$0\$ to \$N-1\$ (which are \$N\$ values). You might as well some from some index \$n_0\$ to \$n_0+N-1\$ (which again are exactly \$N\$ values).