  Divide-and-conquer dividing by a half recurrences, of the form $x_n =a\cdot
x_{\left\lceil{n}/{2}\right\rceil}+a\cdot
x_{\left\lfloor{n}/{2}\right\rfloor}+p(n)$, $n\geq 2$, appear in many areas of
applied mathematics, from the analysis of algorithms to the optimization of
phylogenetic balance indices. The Master Theorems that solve these equations do
not provide the solution's explicit expression, only its big-$\Theta$ order of
growth. In this paper we give an explicit expression (in terms of the binary
decomposition of $n$) for the solution $x_n$ of a recurrence of this form, with
given initial condition $x_1$, when the independent term $p(n)$ is a polynomial
in $\lceil{n}/{2}\rceil$ and $\lfloor{n}/{2}\rfloor$.
