Algebra or Number Theory?

Given that

\[m^2+2(3^n)=m(2^{n+1}-1)\]

has nonnegative integer solutions \((m_1,n_1),(m_2,n_2),(m_3,n_3),\ldots,(m_i,n_i)\), find the value of

\[(m_1+m_2+m_3+\ldots+m_i)+(n_1+n_2+n_3+\ldots+n_i).\]

Details and Assumptions

This problem is not entirely original.

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