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

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.


Introduction
The Fibonacci sequence is named after Leonardo of Pisa To find Fn for a general positive integer n, we hope that we can see a pattern in the sequence of numbers already found. A sharp eye can now detect that any number in the sequence is always the sum of the two numbers preceding it. That is, 2 1 n n n F F F Fibonacci series is helix like identity. It converges to golden ratio, we can show its existence in spiral shells but its elements never construct volumetric object. 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.

Row Matrix Building for Fibonacci's Elements
Difference method In which odd rows numbers are Fibonacci series numbers and even rows numbers are difference of two consecutive odd rows numbers.
In which odd rows numbers are Fibonacci series numbers and even rows numbers are difference of two consecutive odd rows numbers.
is a m × 4 matrix.
In which odd rows numbers are Fibonacci series numbers and even rows numbers are difference of two consecutive odd rows numbers.

Addition Method
In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.
In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.
In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.
In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.
We do the same again and again we get  From above those diagonal differences remains the extinct of Fibonacci's elements.

Difference Chart of Above Series
Diff. 1 Diff.

Difference Parallelogram of Fibonacci Numbers
Above difference parallelogram shows Fibonacci series never vanished, which means it exist everlastingly.

Matrices in Pascal's Elements
be an n × n matrix having Pascal's elements. Where m = n -1. k is an exponent and "a" is variant. Now, Jordan normal matrix of A 1 0 0 0 0 be an y × y matrix having Pascal's elements. Where x = y -1. k is an exponent and "a" is variant. Now, inverse for North-East matrix 3) SE (South-East Pascal's matrix) be an n × n matrix having Pascal's elements. Where m = n -1. k is an exponent and "a" is variant. Now,

4) SW (South-West Pascal's matrix) Let
be an y × y matrix having Pascal's elements. Where x = y -1. k is an exponent and 'a' is variant. Now, inverse for south-west matrix , where n is integer and 0 ≤ a < 6.
Theorem 3: Remainder of arbitrary product of any number of Hex series is always 1 when the product is divided by 6. Proof: We can say above as ( ) ( )

Conclusions
1) Fibonacci series never dies. We can generate so many series like Fibonacci series, they also converge to golden ratio. By this way we find so many golden ratio pairs.
2) Matrix with Pascal elements never vanished at any "n" dimensional matrix calculation. For all arithmetic and matrix operation of matrix with Pascal elements never give up its frame. Here frame means the structure of matrix.
3) Sum of k th elements of hex series gives k 3 and hex series elements form hexagonal only.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.