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

1. Introduction

The Fibonacci sequence is named after Leonardo of Pisa (c. 1170-c. 1250), popularly known as Fibonacci. He wrote a number of books such as Liber Abaci (The Book of Calculating) in 1202, Practica Geometriae (Practical Geometry) in 1220, Flos in 1225, and Liber Quadratorum (The Book of Squares) in 1225. Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding numbers. First few numbers in the series are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 … In India, Fibonacci sequence appeared in Sanskrit prosody (a system of versification). In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables that are 2 units of duration mix with the short (S) syllables that are 1 unit of duration. Counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers—the number of patterns that are m short syllables-long is the Fibonacci number Fm + 1. According to Susantha Goonatilake of Royal Asiatic Society Sri Lanka, the development of the Fibonacci sequence “is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. AD 700), Gopala (c. AD 1135), and Hemchandra (c.AD 1150)”. 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,

${F}_{n+\text{2}}={F}_{n+\text{1}}+{F}_{n}$, for $n=0,1,2,3,\cdots$.

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. Fibonacci series elements construct Area only. Pascal triangle elements (Binomial series elements) construct area, volume and volumetric objects but whatever be it remains its identity which means, if we constructed a matrix with Pascal triangle elements, which would be a square matrix, its kth power or its inverse might have the same identity of Pascal triangle elements, hex series having different numbers, but all numbers will be derived by triangular series numbers.

The Fibonacci Numbers are also applied in Pascal’s Triangle. Entry is sum of the two numbers either side of it, but in the row above. Diagonal sums in Pascal’s Triangle are the Fibonacci numbers. We are getting some ideas from (  Jeffrey R. Chasnov (2016-19) - Fibonacci Numbers and the Golden ratio - Lecture Notes for Course - The Hong Kong University of Science and Technology, Department of Mathematics, Clear Water Bay, Kowloon - Hong Kong). We know Fibonacci Numbers and the Golden ratio (  Tom Davis, Exploring Pascal’s Triangle- tomrdavis@earthlink.net http://www.geometer.org/mathcircles, January 1, 2010; Relation between Pascal’s triangle and Fibonacci’s numbers;  Balasubramani Prema Rangasamy - Some extensions on numbers - Advances in Pure Mathematics, 2019, 9, 944-958. Difference table and  https://en.wikipedia.org/w/index.php?title=Golden_ratio&oldid=83746951"8) We know more about Fibonacci’s elements, Pascal’s elements, Hex numbers and Golden ratio. The Golden Section represented by the Greek letter Phi (φ) = 1.6180339887.

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.

2. Row Matrix Building for Fibonacci’s Elements

Difference method

$\left[\begin{array}{cc}1& 1\\ 1& 2\\ 2& 3\\ 3& 5\\ 5& 8\\ 8& 13\\ 13& 21\\ ⋮& ⋮\end{array}\right]$ is a m × 2 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are difference of two consecutive odd rows numbers.

$\left[\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 5& 8\\ 10& 16& 26\\ 13& 21& 34\\ 42& 68& 110\\ 55& 89& 144\\ ⋮& ⋮& ⋮\end{array}\right]$ is a m × 3 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are difference of two consecutive odd rows numbers.

$\left[\begin{array}{cccc}1& 1& 2& 3\\ 4& 7& 11& 18\\ 5& 8& 13& 21\\ 29& 47& 76& 123\\ 34& 55& 89& 144\\ 199& 322& 521& 843\\ 233& 377& 610& 987\\ ⋮& ⋮& ⋮& ⋮\end{array}\right]$ 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.

$\left[\begin{array}{ccccc}1& 1& 2& 3& 5\\ 7& 12& 19& 31& 50\\ 8& 13& 21& 34& 55\\ 81& 131& 212& 343& 555\\ 89& 144& 233& 377& 610\\ 898& 1453& 2351& 3804& 6155\\ 987& 1597& 2584& 4181& 6765\\ ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right]$ is a m × 5 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are difference of two consecutive odd rows numbers.

We can generate mth series by

${}^{m}F{}_{n+2}={}^{m}F{}_{n+1}+{}^{m}F{}_{n}$ (1)

where

${}^{m}F{}_{n}={}^{1}F{}_{m+n}-{}^{1}F{}_{n}$ and m is an mth Fibonacci’s series. (2)

where ${}^{m}F{}_{n}$ an nth element of a mth Fibonacci series, ${}^{1}F{}_{n}$ is nth element of a 1st Fibonacci series and ${}^{1}F{}_{m+n}$ is m + nth element of a 1st Fibonacci series.

1st series elements are known as Fibonacci numbers.

4th series elements are known as Lucas numbers.

Axiom 1: All the above series are converges to Golden ratio.

$\left[\begin{array}{cc}1& 1\\ 3& 4\\ 2& 3\\ 7& 11\\ 5& 8\\ 18& 29\\ 13& 21\\ ⋮& ⋮\end{array}\right]$ is a m × 2 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.

$\left[\begin{array}{ccc}1& 1& 2\\ 4& 6& 10\\ 3& 5& 8\\ 16& 26& 42\\ 13& 21& 34\\ 68& 110& 178\\ 55& 89& 144\\ ⋮& ⋮& ⋮\end{array}\right]$ is a m × 3 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.

$\left[\begin{array}{cccc}1& 1& 2& 3\\ 6& 9& 15& 24\\ 5& 8& 13& 21\\ 39& 63& 102& 165\\ 34& 55& 89& 144\\ 267& 432& 699& 1131\\ 233& 377& 610& 987\\ ⋮& ⋮& ⋮& ⋮\end{array}\right]$ is a m × 4 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.

$\left[\begin{array}{ccccc}1& 1& 2& 3& 5\\ 9& 14& 23& 37& 60\\ 8& 13& 21& 34& 55\\ 97& 157& 254& 411& 665\\ 89& 144& 233& 377& 610\\ 1076& 1741& 2817& 4558& 7375\\ 987& 1597& 2584& 4181& 6765\\ ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right]$ is a m × 5 matrix.

In which odd rows numbers are Fibonacci series numbers and even rows numbers are addition of two consecutive odd rows numbers.

We can generate kth series by

${}^{k}F{}_{k+2}={}^{k}F{}_{n+1}+{}^{k}F{}_{n}$ (3)

and

${}^{k}F{}_{n}={}^{1}F{}_{k+n}+{}^{1}F{}_{n}$ (4)

where ${}^{k}F{}_{n}$ an nth element of a kth Fibonacci series, ${}^{1}F{}_{n}$ is nth element of a 1st Fibonacci series and ${}^{1}F{}_{k+n}$ is k + 1th element of a 1st Fibonacci series.

1st series elements are known as Fibonacci numbers.

2nd series elements are known as Lucas numbers.

3rd series elements are known as doubled Fibonacci numbers.

4th series elements are known as tripled Fibonacci numbers.

4. Difference between All Series Diagonal Elements

${}^{1}d{}_{n}={}^{s+1}F{}_{n}-{}^{s}F{}_{n+1}$ (6)

and ${}^{s}d{}_{n}={}^{s}F{}_{n+1}-{}^{s+1}F{}_{n};s\ge 2$ (7)

where ${}^{s}d{}_{n}$ is nth element of a sth different series, ${}^{s+1}F{}_{n}$ is nth element of a s + 1th Fibonacci series and ${}^{s}F{}_{n+1}$ is n + 1th element of a sth Fibonacci series.

From above those diagonal differences remains the extinct of Fibonacci’s elements.

5. Difference Chart of Above Series

From the above we chart,

Diff tth series:

${}^{t+2}D{}_{n}={}^{t+1}D{}_{n}+{}^{t}D{}_{n}$ (9)

where

${}^{t}D{}_{n}={}^{t}F{}_{n+1}-{}^{t}F{}_{n}$ (10)

and ${}^{t+1}D{}_{n}={}^{t+1}F{}_{n+1}-{}^{t+1}F{}_{n}$ (11)

where ${}^{t}D{}_{n}$ an nth element of a tth different Fibonacci series, ${}^{1}F{}_{n}$ is nth element of a 1st Fibonacci series and ${}^{1}F{}_{k+n}$ is k + 1th element of a 1st Fibonacci series.

Axiom 3: All the above different series are converges to Golden ratio.

6. Difference Parallelogram of Fibonacci Numbers

Above difference parallelogram shows Fibonacci series never vanished, which means it exist everlastingly.

7. Matrices in Pascal’s Elements

Let

$A=\left[\begin{array}{cccccc}1& & & & & \\ 1& 1& & & & \\ 1& 2& 1& & & \\ 1& 3& 3& 1& & \\ ⋮& ⋮& ⋮& ⋮& \ddots & \\ {}^{m}{C}_{0}& {}^{m}{C}_{1}& {}^{m}{C}_{2}& {}^{m}{C}_{3}& \cdots & {}^{m}{C}_{m}\end{array}\right]$

be an n × n matrix having Pascal’s elements. Where m = n − 1. We called it as Pascal’s matrix.

Now we define Pascal matrix by any variable.

1) NW (North-west Pascal’s matrix)

Let

$A=\left[\begin{array}{cccccc}a& & & & & \\ a& a& & & & \\ a& 2a& a& & & \\ a& 3a& 3a& a& & \\ ⋮& ⋮& ⋮& ⋮& \ddots & \\ {}^{m}{C}_{0}a& {}^{m}{C}_{1}a& {}^{m}{C}_{2}a& {}^{m}{C}_{3}a& \cdots & {}^{m}{C}_{m}a\end{array}\right]$

be an n × n matrix having Pascal’s elements. Where m = n – 1. k is an exponent and “a” is variant.

Now,

${A}^{k}=\left[\begin{array}{cccccc}{k}^{0}{a}^{k}& & & & & \\ {k}^{1}{a}^{k}& {k}^{0}{a}^{k}& & & & \\ {k}^{2}{a}^{k}& 2{k}^{1}{a}^{k}& {k}^{0}{a}^{k}& & & \\ {k}^{3}{a}^{k}& 3{k}^{2}{a}^{k}& 3{k}^{1}{a}^{k}& {k}^{0}{a}^{k}& & \\ ⋮& ⋮& ⋮& ⋮& \ddots & \\ {}^{m}{C}_{0}{k}^{n-1}{a}^{k}& {}^{m}{C}_{1}{k}^{n-2}{a}^{k}& {}^{m}{C}_{2}{k}^{n-3}{a}^{k}& {}^{m}{C}_{3}{k}^{n-4}{a}^{k}& \cdots & {}^{m}{C}_{m}{k}^{0}{a}^{k}\end{array}\right]$

${A}^{-1}=\frac{1}{a}\left[\begin{array}{cccccc}1& & & & & \\ -1& 1& & & & \\ 1& -2& 1& & & \\ -1& 3& -3& 1& & \\ ⋮& ⋮& ⋮& ⋮& \ddots & \\ {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{0}\right]& {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{1}\right]& {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{2}\right]& {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{3}\right]& \cdots & {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{m}\right]\end{array}\right]$.

Jordan normal matrix of A

${J}_{A}=\left[\begin{array}{cccccc}a& 1& 0& 0& 0& 0\\ 0& a& 1& 0& 0& 0\\ 0& 0& a& 1& 0& 0\\ 0& 0& 0& a& 1& 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & 1\\ 0& 0& 0& 0& \cdots & a\end{array}\right]$

2) NE (North-East Pascal’s matrix)

Let

$A=\left[\begin{array}{cccccc}& & & & & a\\ & & & & a& a\\ & & & a& 2a& a\\ & & a& 3a& 3a& a\\ & ⋰& ⋮& ⋮& ⋮& ⋮\\ {}^{x}{C}_{x}a& \cdots & {}^{x}{C}_{3}a& {}^{x}{C}_{2}a& {}^{x}{C}_{1}a& {}^{x}{C}_{0}a\end{array}\right]$

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

${A}^{-1}=\frac{1}{a}\left[\begin{array}{cccccc}{\left(-1\right)}^{y-1}\left[{}^{x}{C}_{0}\right]& {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{1}\right]& {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{2}\right]& {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{3}\right]& \cdots & {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{x}\right]\\ ⋮& ⋮& ⋮& ⋮& ⋰& \\ -1& 3& -3& 1& & \\ 1& -2& 1& & & \\ -1& 1& & & & \\ 1& & & & & \end{array}\right]$

3) SE (South-East Pascal’s matrix)

Let

$A=\left[\begin{array}{cccccc}{}^{m}{C}_{m}a& \cdots & {}^{m}{C}_{3}a& {}^{m}{C}_{2}a& {}^{m}{C}_{1}a& {}^{m}{C}_{0}a\\ & \ddots & ⋮& ⋮& ⋮& ⋮\\ & & a& 3a& 3a& a\\ & & & a& 2a& a\\ & & & & a& a\\ & & & & & a\end{array}\right]$

be an n × n matrix having Pascal’s elements. Where m = n – 1. k is an exponent and “a” is variant.

Now,

${A}^{-1}=\frac{1}{a}\left[\begin{array}{cccccc}{\left(-1\right)}^{n-1}\left[{}^{m}{C}_{m}\right]& & {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{3}\right]& {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{2}\right]& {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{1}\right]& {\left(-1\right)}^{n-1}\left[{}^{m}{C}_{0}\right]\\ & \ddots & ⋮& ⋮& ⋮& ⋮\\ & & 1& -3& 3& -1\\ & & & 1& -2& 1\\ & & & & 1& -1\\ & & & & & 1\end{array}\right]$.

4) SW (South-West Pascal’s matrix)

Let

$A=\left[\begin{array}{cccccc}{}^{x}{C}_{0}a& {}^{x}{C}_{1}a& {}^{x}{C}_{2}a& {}^{x}{C}_{3}a& \cdots & {}^{x}{C}_{x}a\\ ⋮& ⋮& ⋮& ⋮& ⋰& \\ a& 3a& 3a& a& & \\ a& 2a& a& & & \\ a& a& & & & \\ a& & & & & \end{array}\right]$

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

${A}^{-1}=\frac{1}{a}\left[\begin{array}{cccccc}& & & & & 1\\ & & & & 1& -1\\ & & & 1& -2& 1\\ & & 1& -3& 3& -1\\ & ⋰& ⋮& ⋮& ⋮& ⋮\\ {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{x}\right]& \cdots & {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{3}\right]& {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{2}\right]& {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{1}\right]& {\left(-1\right)}^{y-1}\left[{}^{x}{C}_{0}\right]\end{array}\right]$

Hex numbers:

Let h be any hex number. We know mod 6 of any h is equal to 1.

Mod 6 of h1 ≡ 1; Mod 6 of h2 ≡ 1; $\cdots$ ; Mod 6 of hk ≡ 1;

Theorem 1: Difference between any two elements of Hex numbers is fully divided by 6.

Theorem 2: $\underset{k=0}{\overset{\infty }{\sum }}{H}_{6n±k}\equiv a\mathrm{mod}6$, 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:

$R=\frac{\underset{i}{\prod }{H}_{i}}{6}=1$.

We can say above as

$R\left({h}_{1}×{h}_{2}×\cdots ×{h}_{k}\right)÷6\equiv \left(1×1×\cdots ×1\right)\mathrm{mod}6=1$.

Theorem 4: $\underset{o}{\overset{}{\sum }}{H}_{k}\mathrm{mod}\left(6\right)\equiv \underset{E}{\overset{}{\sum }}{H}_{k}\left(\mathrm{mod}6\right)|k\in Z$

Theorem 5: $\underset{k=0}{\overset{\infty }{\sum }}{H}_{k}={k}^{3}$, where Hk is Hex series elements.

Matrices of Hex numbers

Let we see the relation between hex numbers in matrix

1) Let $A=\left[\begin{array}{cc}1& h+1\\ 3h+1& 6h+1\end{array}\right]$ be a 2 × 2 matrix which elements are hex numbers (where h = 6) then $|A|=|\begin{array}{cc}1& h+1\\ 3h+1& 6h+1\end{array}|=6h+1-3{h}^{2}-4h+1=-3{h}^{2}+2h$

2) Let $A=\left[\begin{array}{cc}h+1& 3h+1\\ 6h+1& 10h+1\end{array}\right]$ be a 2 × 2 matrix which elements are hex numbers (where h = 6) then $|A|=|\begin{array}{cc}h+1& 3h+1\\ 6h+1& 10h+1\end{array}|=-8{h}^{2}+2h$

By above way we get, $-15{h}^{2}+2h$ ; $-24{h}^{2}+2h$ ; $\cdots$ ; $-n\left(n+2\right){h}^{2}+2h$

1) Let we construct a difference triangle about above determinants

a) Let $A=\left[\begin{array}{ccc}1& h+1& 3h+1\\ 6h+1& 10h+1& 15h+1\\ 21h+1& 28h+1& 36h+1\end{array}\right]$ be a 3 × 3 matrix which elements are hex numbers then $|A|=|\begin{array}{ccc}1& h+1& 3h+1\\ 6h+1& 10h+1& 15h+1\\ 21h+1& 28h+1& 36h+1\end{array}|=-27{h}^{3}$

b) $|A|=|\begin{array}{ccc}h+1& 3h+1& 6h+1\\ 10h+1& 15h+1& 21h+1\\ 28h+1& 36h+1& 45h+1\end{array}|=-27{h}^{3}$

2) Let we construct a difference triangle about above determinants

a) Let $A=\left[\begin{array}{cccc}1& h+1& 3h+1& 6h+1\\ 10h+1& 15h+1& 21h+1& 28h+1\\ 36h+1& 45h+1& 55h+1& 66h+1\\ 78h+1& 91h+1& 105h+1& 120h+1\end{array}\right]$ be a 4 × 4 matrix which elements are hex numbers then $|A|=|\begin{array}{cccc}1& h+1& 3h+1& 6h+1\\ 10h+1& 15h+1& 21h+1& 28h+1\\ 36h+1& 45h+1& 55h+1& 66h+1\\ 78h+1& 91h+1& 105h+1& 120h+1\end{array}|=0$

b) $|A|=|\begin{array}{cccc}h+1& 3h+1& 6h+1& 10h+1\\ 15h+1& 21h+1& 28h+1& 36h+1\\ 45h+1& 55h+1& 66h+1& 78h+1\\ 91h+1& 105h+1& 120h+1& 136h+1\end{array}|=0$

3) Let we construct a difference triangle about above determinants

From the above we can state:

a) Determinants of 2 × 2 matrix with hex series elements vanished at 2nd difference.

b) Determinant of 3 × 3 matrix with hex series elements vanished at 0th difference.

c) Determinant of 4 × 4 matrix with hex series elements and above are 0.

Which means hex series elements are forming hexagonal only.

8. 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 kth elements of hex series gives k3and hex series elements form hexagonal only.

Cite this paper: Rangasamy, B. (2020) Non Degeneration of Fibonacci Series, Pascal’s Elements and Hex Series. Advances in Pure Mathematics, 10, 393-404. doi: 10.4236/apm.2020.107024.
References

   Chasnov, J.R. (2016) Fibonacci Numbers and the Golden Ratio. Lecture Notes for Course, The Hong Kong University of Science and Technology, Department of Mathematics, Clear Water Bay, Kowloon, Hong Kong.

   Davis, T. (2010) Exploring Pascal’s Triangle.
http://www.geometer.org/mathcircles

   Rangasamy, B.P. (2019) Some Extensions on Numbers. Advances in Pure Mathematics, 9, 944-958.
https://doi.org/10.4236/apm.2019.911047

   https://en.wikipedia.org/w/index.php?title=Golden_ratio&oldid=83746951"8

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