Short note on additive sequences and on recursive processes

TL;DR

This paper explores generalized recursive and additive sequences, revealing new ratio limits, identities, and behaviors in dynamical systems, chaos, and formal grammars, highlighting their mathematical properties and connections.

Contribution

It introduces novel methods for generalizing recursive processes, uncovering new ratio limits, identities, and behaviors across various mathematical and dynamical systems.

Findings

Existence of dense ratio limits in additive sequences.

Identification of properties of special recursive sequences.

Observation of complex orbit behaviors in dynamical systems.

Abstract

Simple methods permit to generalize the concepts of iteration and of recursive processes. We shall see briefly on several examples what these methods generate. In additive sequences, we shall encounter not only the golden or the silver ratio, but a dense set of ratio limits that corresponds to an infinity of conceivable recursive additive rules. We shall show that some of these limits have nice properties. Identities involving Fibonacci and Lucas sequences will be viewed as special cases of more general identities. We shall show that some properties of the Pascal Triangle belong also to other similar objects. In Dynamical Systems and Chaos Theory we shall encounter weird orbits, whose order is higher than the number of its distinct elements and, beyond the chaos point, a rather unexpected belated convergence to 0, after a pseudo chaotic behaviour during as many terms as one may wish. Time and again, we shall find here the Feigenbaum constant. In Formal Grammars we shall see that recursive rules applied to concatenation are sometimes equivalent to formal grammars although generally more restrictive.