Generalized Fibonacci numbers, cosmological analogies, and an invariant

TL;DR

This paper explores continuous generalizations of Fibonacci numbers through differential equations analogous to cosmological models, presenting their mathematical formulations and an invariant property.

Contribution

It introduces a novel continuous extension of Fibonacci sequences, linking them to cosmological equations and providing their Lagrangian, Hamiltonian, and invariant formulations.

Findings

Fibonacci generalizations satisfy specific ODEs similar to Friedmann equations

The paper derives Lagrangian and Hamiltonian formulations for these sequences

An invariant of the generalized Fibonacci sequence is identified

Abstract

Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues of the Friedmann equation describing 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.