F0 does not affect formant frequency, for a slightly obscure reason. When you look at a spectral cross section, you see the amplitudes of individual harmonics (determined via a Fourier transform). The peaks occur pretty much at multiples of the fundamental. The harmonic with the highest amplitude in the area where you expect a formant is not necessarily at the formant's frequency: the actual formant's peak is usually between two harmonics.
A formant frequency is computed using LPC analysis, usually, which gives you a general picture of the resonances in the vocal tract, independent of fundamental frequency. The underlying theory is known as the source-filter theory, where a given glottal source (pitch and voice quality characteristics) is "shaped" by the LPC coefficients to give an actual waveform.
In other words, the mathematics of formant analysis guarantees that pitch and formant frequency are independent. However: the situation you describe in the formulas does not correspond to a real situation, an you will not find a vowel whose formant's are doubled when pitch is doubled. 3000 Hz is not a realistic F2, nor is 1600 a realistic F1