Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant

Valerio Faraoni,Farah Atieh

doi:10.3390/sym13020200

Abstract

Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues of the Friedmann equation describing a spatially homogeneous and isotropic cosmology in general relativity. These analogies are presented together with their Lagrangian and Hamiltonian formulations and with an invariant of the Fibonacci sequence.

Highlights

Fibonacci, known as Leonardo Pisano or Leonardo Bonacci, introduced Hindu–Arabic numerals to Europe with his book Liber Abaci in 1202 [1]
We focus on F(e,o) ( x ), which admit analogies with relativistic cosmology, while no such analogy exists for F ( x )
Of which F(e) and F(o) are two linearly independent solutions. This equation describes the one-dimensional motion of a particle of position F in the inverted harmonic oscillator potential V ( F ) = −kF2 /2, which is used as an example of an unstable mechanical system

Summary

Introduction

Arabic numerals to Europe with his book Liber Abaci in 1202 [1]. He posed and solved a well-known problem involving the growth of a population of rabbits in idealized situations. The ratio of two consecutive terms Fn+1 /Fn approaches the golden ratio. Reproduces part of the Fibonacci numbers, Fn = F(e) (n), for even x = n ∈ N, while. Reproduces the other Fibonacci numbers for odd x = n ∈ N. Of which F(e) and F(o) are two linearly independent solutions In physics, this equation describes the one-dimensional motion of a particle of position F in the inverted harmonic oscillator potential V ( F ) = −kF2 /2 (with K = 2 ln φ), which is used as an example of an unstable mechanical system (this property corresponds to the fact that the Fibonacci numbers Fn increase without bound as n → ∞). We briefly recall the basics of spatially homogeneous and isotropic cosmology in general relativity, and we present the formal analogy with.

FLRW Cosmology

The Cosmological Analogy

Lagrangian and Hamiltonian

Conclusions