# If a time-series has odd number of samples does it have no energy at Nyquist frequency?

Suppose I have real time series A with n samples and time-spacing dt and I want to analyze its frequency content.

Af = fft(A)


If dt=1 Nyquist frequency is 0.5. According to both Numpy and Matlab manuals, the frequency vector is defined as

f = [0, 1, ...,   n/2-1,     -n/2, ..., -1] / (dt*n)   if n is even

f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (dt*n)   if n is odd

If n=9 (odd), f= [-0.44444 -0.33333 -0.22222 -0.11111  0.  0.11111 0.22222  0.33333  0.44444]
If n=8 (even), f= [-0.5   -0.375 -0.25  -0.125  0.     0.125  0.25   0.375]


Note the maximum frequency for odd number of samples is 0.44444 Hz not 0.5 Hz. Does this mean a time series with odd number of samples has no energy at Nyquist frequency ?

$$E_{Nyq}=\left|\sum_{n=0}^{N-1}x[n](-1)^n\right|^2\tag{1}$$
where $$N$$ is the length of the sequence, no matter if it's even or odd.