Question $3$

For

$$

in

$$

$0$

n in N

$0$

$$

let

$$

$($

$$

$)$

w$($n$)$ denote the number of

$1$

$1$s in the binary representation of

$$

n$.$ For example,

$$

$($

$9$

$)$

$=$

$2$

w$(9)=2,$ since $$9$$ is

$1001$

$1001$ in binary. Try to find a closed formula for

$$

$($

$$

$)$

g$($n$)$ in terms of

$$

n and

$$

$($

$$

$)$

w$($n$).$ If you succeed, the following question will be very easy.