TITLE:
Non Degeneration of Fibonacci Series, Pascal’s Elements and Hex Series
AUTHORS:
Balasubramani Prema Rangasamy
KEYWORDS:
Fibonacci Series, Lucas Series, Golden Ratio, Various Type of Fibonacci Series Generated by Matrices, Matrix Operations on Pascal’s Elements and Hex Numbers
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.10 No.7,
July
21,
2020
ABSTRACT: 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.