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

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 $\frac{1}{{\varphi }^n}$, 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.

9 pages, 1 figure, Fibonacci Quaterly