Non Degeneration of Fibonacci Series, Pascal’s Elements and Hex Series

• 作者: Balasubramani Prema Rangasamy
• 来源期刊: 理论数学进展(英文)
• 学科: 数学
• 关键词: Fibonacci Series Lucas Series Golden Ratio Various Type of Fibonacci Series Generated by Matrices Matrix Operations on Pascal’s Elements and Hex Numbers
• 年，卷（期）: 2020,（7）
• 页码范围: 393-404
• 页数: 12页
• 分类号: O15

摘要

Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.