P_n = 2 + 4 + \dots + 2\left\lfloor\frac{n}2\right\rfloor = 2 \left(1 + 2 + \dots + \left\lfloor\frac{n}2\right\rfloor\right) = 2S_{\left\lfloor\frac{n}2\right\rfloor}