How could we get \(0,1,0,1,0,1,\ldots\text{?}\) Start with the previous sequence and shift it over by 1. But how do you do this? To see how shifting works, let's first try to get the generating function for the sequence \(0, 1, 3, 9, 27, \ldots\text{.}\) We know that \(\frac{1}{1-3x} = 1 + 3x + 9x^2 + 27x^3 + \cdots\text{.}\) To get the zero out front, we need the generating series to look like \(x + 3x^2 + 9x^3 + 27x^4+ \cdots\) (so there is no constant term). Multiplying by \(x\) has this effect. So the generating function for \(0, 1, 3, 9, 27, \ldots\) is \(\frac{x}{1-3x}\text{.}\) This will also work to get the generating function for \(0,1,0,1,0,1,\ldots\text{:}\)