2017/07/28 by Merve Özvatan, Oktay K. Pashaev, Özvatan, Merve +1 · 1 citation
Materials Science · Medicine · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Biofield Effects and Biophysics #FOS: Mathematics #History and Overview (math.HO) #Number Theory (math.NT) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.1707.09151
openalex publication_date 2017/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∞=(2)/(5 π) ln(φ).