From Pascal’s Triangle to a Mathematical Frontier: Uncovering Hidden Patterns in the Oscillations of Generalized Golden Ratio

This note demonstrates how, using commonly available computer applications, allows for the discovery of patterns that structure oscillations of generalized golden ratios formed by the roots of the so-called Fibonacci-like polynomials. It is shown how the universality of oscillations associated with the smallest root, regardless of the length of the string of the ratios, can be established and formulated in combinatorial terms. In addition, the use of circular diagrams informed by computational experiments in the case of the strings of odd number lengths made it possible to recognize patterns associated with the largest root.