Waring decompositions of special binomials

We determine the Waring rank of homogeneous polynomials of the form $x^ky^kz^k + \ell^{3k}$ where $\ell$ is a linear form. The result is based on the study of the Hilbert function and the resolution of special configurations of points in $\mathbb{P}^2$. As a byproduct of our result, we show that the monomial $x^ky^kz^k$ does not have irredundant decompositions of length $(k+1)^2 +1$.