  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.
