Solvable difference equations similar to the Newton-Raphson iteration for algebraic equations

It is known that difference equations generated as the Newton-Raphson iteration for quadratic equations are solvable in closed form, and the solution can be constructed from linear three-term recurrence relations with constant coefficients. We show that the same construction for four-term recurrence relations gives the solution to the initial value problem of difference equations similar to the Newton-Raphson iteration for cubic equations. In many cases, the solution converges to a root of the cubic equation and the convergence rate is quadratic.