Fast Fourier Transform

[Eddie W. Shore]

I hope you can help me. I am trying to understand how the fast Fourier transform is calculated (FFT).

Sample problem:
n_0 = 0.54
n_1 = 0.66
n_2 = 0.52
where N = 3

The formula to the FFT (I think) is:
X_k = Σ( x_n * e^(-i * 2 * π * k * n / N) ) for n = 0 to N-1

Using the formula above I get:
X_0 = 1.62
X_1 = 0
X_2 = 0

But the fft function on the HP Prime returns:
1.72
-0.05 - 0.12124355653i
-0.05 + 0.12124355653i

However Wolfram Alpha returns:
0.993042
-0.0288675 + 0.07i
-0.0288675 - 0.07i

I am confused. Are there different fast Fourier transforms or am I missing something obvious? I want to understand the basic calculation before I attempt to understand the Tukey and Cooley algorithm. Any help and insight is appreciated. Thanks!