From Fibonacci Sequence to the Golden Ratio

Abstract

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

Hecke groups, linear recurrences, and Kepler limits

“…It is well known, of course, that the consecutive ratios F n /F n−1 of Fibonacci numbers converge to φ. More generally, the limit of the ratios of consecutive elements of a linear recurrence L, when it exists, is called by Fiorenza and Vincenzi the Kepler limit of L. Certain roots of other polynomials than x 2 − x − 1 are also Kepler limits [19,20], so we are led to consider the possibility that the G 5 phenomenon generalizes; our observations tend to confirm this guess. Section 2 describes what we found out about G 5 , section 3 describes less detailed observations for G k , 5 ≤ k ≤ 33, and the final section provides some detail about our numerical experiments; documentation in the form of Mathematica notebooks is on our ResearchGate pages [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].…”

On ̄h-Jacobsthal and ̄h-Jacobsthal–Lucas sequences, and related quaternions

“…n is definitively different from 0 and it has Kepler limit if and only if |h| = | − 1| = 1. It is well known that in such case the Kepler limit is exactly the root of maximum modulus (see for example [12], and references therein), and this shows the condition 'i'.…”

Child Drawing Development Optimization Algorithm Based on Child’s Cognitive Development

“…Due to its unique properties, it has been used in many fields, such as architecture, art, and design, applied in outstanding works of aircraft, sculptures, paintings, and architecture. The golden ratio can be found in the natural world in many forms, such as the body proportion of living beings, insects, and growth patterns of plants [30]. Mathematical series and geometrical patterns are used to represent the Golden Section (GS).…”