사인 제곱 수열 합

$\displaystyle a _{1} ^{2} = \frac{1}{2}$, $a _{2} ^{2} = 1$, $\displaystyle a _{3} ^{2} = \frac{1}{2}$, $a _{4} ^{2} = 0$, $\displaystyle a _{5} ^{2} = \frac{1}{2}$, $a _{6} ^{2} = 1$, $\displaystyle a _{7} ^{2} = \frac{1}{2}$, $a _{8} ^{2} = 0$, $\cdots$ $\displaystyle \sum\limits _{n = 1} ^{32} na _{n} ^{2} = \frac{1}{2} \times \frac{16 \times 32}{2} + 1 \times \frac{8 \times 32}{2} = 256$