On the equation \(f(n^2+Dm^2+k)=f(n)^2+Df(m)^2+k\)

Abstract. We give all solutions of the equation $$f(n^2+Dm^2+k)=f(n)^2+Df(m)^2+k,$$ where \(D, k\in\mathbb{N}\) are given and \(f\) is an arbitrary complex valued function defined on \(\mathbb{N}\).