The Discrete Fourier Transform of an N-point sequence $\left\{x_n\right\} := x_0, x_1, ..., x_{N-1}$ is given by
$$ X(k) = \sum^{N-1}_{n=0}x(n)e^{-j \frac{2\pi}{N}nk} $$
$$ x(n) = \frac{1}{N}\sum^{N-1}_{k=0}X(k)e^{j \frac{2\pi}{N}nk} $$
Compute the DFT of x(n) = [0,1,2,3]
Answer $X(k) = [6, -2 + 2j, -2, -2 - 2j]$
$$ X(k) = \sum^1_{n=0}x(n)W_2^{kn} = x(0)W^0_2 + x(1)W_2^k $$
where $W_2 = e^{-j\pi}$
$$ X(k) = \sum^3_{n=0}x(n)W_4^{kn} $$
where $W_4 = e^{-j\frac{\pi}{2}}$