Generalized Fibonacci Sequences and Binet-Fibonacci Curves

TL;DR

This paper explores various generalizations of Fibonacci sequences, extends Binet's formula to real numbers, and analyzes the geometric properties of the resulting Binet-Fibonacci curves in the complex plane.

Contribution

It introduces new generalized Fibonacci sequences, extends Binet's formula to real numbers, and characterizes the geometric behavior of Binet-Fibonacci curves.

Findings

Binet-Fibonacci curves oscillate with exponentially vanishing amplitude for positive numbers.

For negative numbers, the curves form Binet Fibonacci spirals.

The area under the curves converges to a finite value as n approaches infinity.

Abstract

We have studied several generalizations of Fibonacci sequences as the sequences with arbitrary initial values, the addition of two and more Fibonacci subsequences and Fibonacci polynomials with arbitrary bases. For Fibonacci numbers with congruent indices we derived general formula in terms of generalized Fibonacci polynomials and Lucas numbers. By extending Binet formula to arbitrary real numbers we constructed Binet-Fibonacci curve in complex plane. For positive numbers the curve is oscillating with exponentially vanishing amplitude, while for negative numbers it becomes Binet Fibonacci spiral. Comparison with the nautilus curve shows quite similar behavior. Areas under the curve and curvature characteristics are calculated, as well as the asymptotic of relative characteristics. Asymptotically for n going to infinity the region become, infinitely wide with infinitesimally small height so that the area has finite value $A_{\infty}=\frac{2}{5 \pi} ln(\varphi)$.