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Let $G(1) = 0, \ G(2) = 1$, $G(2n+1) = 2 + G(n) + G(n+1)$ and $G(2n) = 1 + G(n), \ \ n \geq 1$

Find $G(n) $

P.S: This is little problem in my problem. I tried to solve by using generating function, but I can not. Can anyone help me. Thanks in advance.

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    $H_{ m }(p_{ m },p_{ m-1 },...,p_{ 1 })=G(2^{ p_{ m } }(2^{ p_{ m-1 } }(...(2^{ p_{ 1 } }+1)...)+1)+1)$, $\left\{\begin{matrix} H_{ n }=(p_{ n }+1)({p_{ n }+2})/2+p_{ n }H_{n-1}+{ H }'_{ n -1} \\ { H }'_{ n }=(p_n+1)p_n/2+(p_n-1)H_{n-1}+{ H }'_{ n -1} \end{matrix}\right.$2012-11-22

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