\[ \large \sum_{n=1}^\infty \dfrac{N_1(n)+ N_0(n)}{n(2n+1)} \]

Let \(N_0(x) \) and \(N_1(x) \) denote the number of zeros and ones, respectively, when \(x\) is written in base two.

If the value of the series above is equal to \( a\gamma \), where \(\gamma\) denotes the Euler-Mascheroni constant, find \(a\).

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