there I just want to know the summation formula for that summation. $$ \sum\limits_{i=0}^n 2^i(n-i) $$
2026-03-28 20:13:39.1774728819
Summation formula for bit changes. $ \sum\limits_{i=0}^n 2^i(n-i) $
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See that $$\sum\limits_{j=0}^n 2^j(n-j)\overset{i=n-j}{=}\sum\limits_{i=0}^n i2^{n-i} =2^{n-1}\sum\limits_{i=0}^n i\frac{1}{2^{i-1}} \\=2^{n-1} \frac{d}{dx}\left(\sum\limits_{i=0}^n x^i\right)\Bigg|_{x=1/2} =\color{blue}{2^{n+1}-n-2}$$