Summary: The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the second-order nonlinear difference equation \[ \Delta^2x(n)+\frac{1}{n(n+1)}f(x(n))=0, \] where \(f(x)\) is continuous on \(\mathbb R\) and satisfies the signum condition \(xf(x)>0\) if \(x\neq 0\). The obtained results are best possible in a certain sense. Proof is given by means of the Riccati technique and phase plane analysis of a system. A discrete version of the Riemann-Weber generalization of Euler-Cauchy differential equation plays an important role in proving our results.