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Let $\mathcal{H}_n$ denote the Hamming weight of some nonnegative integer $n$, then \[ (-1)^{\mathcal{H}_n} = \displaystyle \sum_{k=0}^{2L_n} \,\left(\frac{\sqrt{3}}{2}\right)^{\!2 L_n - k - 1} \,\,\sum_{m=0}^{\left\lceil 2^{k-1}-1\right\rceil} \displaystyle \sin\left(\frac{(6m+1)(2n+1)-(-2)^{k-1}} {3\cdot 2^k} \pi\right) \prod_{i=0}^k \sin\left(\frac{(6m+1)(-2)^i}{3\cdot 2^k}\pi\right) \] with $L_n=\left\lfloor\log_4(n+1)+1\right\rfloor$.
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Tue, 28 Feb 2023 17:25 GMT