# Different Euler Bases

Calculus Level 5

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

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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