Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple andWolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period p among the roots responsible for the cycles of periods 2kp (period-doubling) and kp (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics.