The spectrum of the square wave contains just the nonzero values only at the odd multiple of its frequency(0 ,f0 ,3*f0 ).The intuition behind it is the spectrum value/Fourier series coefficient at given frequency is the measure of correlation(multiply and add) between the signal and sinusoidal signal of given frequency.Consider and square wave(anti-symmetric around zeros) of some frequency shown in the figure.The value of spectrum at given  frequency is given by multiplying the sine/cosine of that frequency and adding(integrating).As you can  see in the figure that the sine/cosine wave of frequency that is even multiple of the square wave,the positive and negative parts of equal area and cancel each other and the correlation is non zero for the odd multiple of the the square wave frequency…-Achuth