2023/08/26 by Schütz, Clemens, Kelly, Kristian
#11B39 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2308.13878
The Fibonacci sequence, Fn = Fn - 1 + Fn - 2, and its counterpart for n < 0, the negaFibonacci sequence, F-n = (-1)n + 1 ⋅ Fn, 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 ωn = ωn - 2 - ωn - 1 with ω1, ω2 ∈ ℂ chosen at will. We will partition this kind into natural complete and complex complete negaFibonacci-esque sequences.We will prove that (1)/(ϕn), 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.