Analytic and Geometric Representations of the Generalized n-anacci Constants

TL;DR

This paper explores the generalization of n-anacci constants through linear recurrences, providing analytic formulas and geometric representations that illustrate their properties and convergence behavior.

Contribution

It introduces analytic expressions and geometric models for generalized n-anacci constants, extending understanding of their limits and geometric interpretations.

Findings

Ratio limits form a strictly increasing sequence converging to p+1.

Geometric representations using convex sets provide intuitive understanding.

Explicit formulas and constructions for various geometric shapes are developed.

Abstract

We study generalizations of the sequence of the n-anacci constants that consist of the ratio limits generated by linear recurrences of an arbitrary order n with equal positive weights p. We derive the analytic representation of these ratio limits and prove that, for a fixed p, the ratio limits form a strictly increasing sequence converging to p+1. We also construct uniform geometric representations of the sequence of the n-anacci constants and generalizations thereof by using dilations of compact convex sets with varying dimensions n. We show that, if the collections of the sets consist of n-balls, n-cubes, n-cones, n-pyramids, etc., then the representations of the generalized n-anacci constants have clear geometric interpretations.