On negaFibonacci-esque Sequences and Their Relation to the Golden Ratio

TL;DR

This paper introduces negaFibonacci-esque sequences, explores their properties, especially their relation to the golden ratio, and provides explicit formulas and classifications for these sequences.

Contribution

It defines a new class of sequences called negaFibonacci-esque sequences, analyzes their structure, and links them to the golden ratio and spiral.

Findings

The principal negaFibonacci-esque sequence is 1/phi^n.

Explicit formulas for complex complete negaFibonacci-esque sequences are provided.

Connections between these sequences and the golden spiral are established.

Abstract

The Fibonacci sequence, $F_n = F_{n - 1} + F_{n - 2}$, and its counterpart for $n < 0$, the negaFibonacci sequence, $F_{-n} = (-1)^{n + 1} \cdot F_n$, are among the most studied sequences in mathematics. In this paper we will present a new kind of sequence, the negaFibonacci-esque sequences, identified by the property that $\omega_n = \omega_{n - 2} - \omega_{n - 1}$ with $\omega_1, \omega_2 \in \mathbb{C}$ chosen at will. We will partition this kind into natural complete and complex complete negaFibonacci-esque sequences.We will prove that $\frac{1}{\phi^n}$, referred to as the principal negaFibonacci-esque sequence, is not only a complex complete negaFibonacci-esque sequence but also one of the most significant. Furthermore, we will present an explicit formula for all complex complete negaFibonacci-esque sequences constructed with a combination of two negaFibonacci terms. We shall then conclude by connecting these sequences to the golden spiral and ratio.