Why is this? Could you make it work using, say, 2n instead of n? or n^2 instead of n?

The definition of the Fourier series is an infinite series of integer frequency multiples adding up to reproduce any periodic signal, sort of like how the Taylor series can additively reproduce a function.

https://en.wikipedia.org/wiki/Fourier_series

Yes! If you skip every other term you'll be going up by 2n.

I'm a bit rusty but going up by 2n (1n,2n,4n,6n...) gives you the even harmonics, the opposite (1n,3n,5n,7n...) gives you the odd harmonics. Old tube-amps are known for keeping the odd harmonics around, it's a "warmer" sound.

edit: fixed the series

That would be the same as this, actually. (For 2n, leave odd terms blank. Etc.)