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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.

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 + 2 = F n + 1 + F n , for n = 0 , 1 , 2 , 3 , ⋯ .

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 k^{th} 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 ( [

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.

Difference method

[ 1 1 1 2 2 3 3 5 5 8 8 13 13 21 ⋮ ⋮ ] 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.

[ 1 1 2 2 4 6 3 5 8 10 16 26 13 21 34 42 68 110 55 89 144 ⋮ ⋮ ⋮ ] 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.

[ 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 ⋮ ⋮ ⋮ ⋮ ] 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.

[ 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 ⋮ ⋮ ⋮ ⋮ ⋮ ] is a m × 5 matrix.

We can generate m^{th} series by

F m n + 2 = F m n + 1 + F m n (1)

where

F m n = F 1 m + n − F 1 n and m is an m^{th} Fibonacci’s series. (2)

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

1^{st} series elements are known as Fibonacci numbers.

4^{th} series elements are known as Lucas numbers.

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

[ 1 1 3 4 2 3 7 11 5 8 18 29 13 21 ⋮ ⋮ ] 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.

[ 1 1 2 4 6 10 3 5 8 16 26 42 13 21 34 68 110 178 55 89 144 ⋮ ⋮ ⋮ ] 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.

[ 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 ⋮ ⋮ ⋮ ⋮ ] 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.

[ 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 ⋮ ⋮ ⋮ ⋮ ⋮ ] is a m × 5 matrix.

We can generate k^{th} series by

F k k + 2 = F k n + 1 + F k n (3)

and

F k n = F 1 k + n + F 1 n (4)

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

1^{st} series elements are known as Fibonacci numbers.

2^{nd} series elements are known as Lucas numbers.

3^{rd} series elements are known as doubled Fibonacci numbers.

4^{th} series elements are known as tripled Fibonacci numbers.

d 1 n = F s + 1 n − F s n + 1 (6)

and d s n = F s n + 1 − F s + 1 n ; s ≥ 2 (7)

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

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

From the above we chart,

Diff t^{th} series:

D t + 2 n = D t + 1 n + D t n (9)

where

D t n = F t n + 1 − F t n (10)

and D t + 1 n = F t + 1 n + 1 − F t + 1 n (11)

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

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

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

Let

A = [ 1 1 1 1 2 1 1 3 3 1 ⋮ ⋮ ⋮ ⋮ ⋱ m C 0 m C 1 m C 2 m C 3 ⋯ m C m ]

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 = [ a a a a 2 a a a 3 a 3 a a ⋮ ⋮ ⋮ ⋮ ⋱ m C 0 a m C 1 a m C 2 a m C 3 a ⋯ m C m a ]

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

Now,

A k = [ 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 ⋮ ⋮ ⋮ ⋮ ⋱ 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 ⋯ m C m k 0 a k ]

A − 1 = 1 a [ 1 − 1 1 1 − 2 1 − 1 3 − 3 1 ⋮ ⋮ ⋮ ⋮ ⋱ ( − 1 ) n − 1 [ m C 0 ] ( − 1 ) n − 1 [ m C 1 ] ( − 1 ) n − 1 [ m C 2 ] ( − 1 ) n − 1 [ m C 3 ] ⋯ ( − 1 ) n − 1 [ m C m ] ] .

Jordan normal matrix of A

J A = [ a 1 0 0 0 0 0 a 1 0 0 0 0 0 a 1 0 0 0 0 0 a 1 0 ⋮ ⋮ ⋮ ⋮ ⋱ 1 0 0 0 0 ⋯ a ]

2) NE (North-East Pascal’s matrix)

Let

A = [ a a a a 2 a a a 3 a 3 a a ⋰ ⋮ ⋮ ⋮ ⋮ x C x a ⋯ x C 3 a x C 2 a x C 1 a x C 0 a ]

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 = 1 a [ ( − 1 ) y − 1 [ x C 0 ] ( − 1 ) y − 1 [ x C 1 ] ( − 1 ) y − 1 [ x C 2 ] ( − 1 ) y − 1 [ x C 3 ] ⋯ ( − 1 ) y − 1 [ x C x ] ⋮ ⋮ ⋮ ⋮ ⋰ − 1 3 − 3 1 1 − 2 1 − 1 1 1 ]

3) SE (South-East Pascal’s matrix)

Let

A = [ m C m a ⋯ m C 3 a m C 2 a m C 1 a m C 0 a ⋱ ⋮ ⋮ ⋮ ⋮ a 3 a 3 a a a 2 a a a a a ]

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

Now,

A − 1 = 1 a [ ( − 1 ) n − 1 [ m C m ] ( − 1 ) n − 1 [ m C 3 ] ( − 1 ) n − 1 [ m C 2 ] ( − 1 ) n − 1 [ m C 1 ] ( − 1 ) n − 1 [ m C 0 ] ⋱ ⋮ ⋮ ⋮ ⋮ 1 − 3 3 − 1 1 − 2 1 1 − 1 1 ] .

4) SW (South-West Pascal’s matrix)

Let

A = [ x C 0 a x C 1 a x C 2 a x C 3 a ⋯ x C x a ⋮ ⋮ ⋮ ⋮ ⋰ a 3 a 3 a a a 2 a a a a a ]

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 = 1 a [ 1 1 − 1 1 − 2 1 1 − 3 3 − 1 ⋰ ⋮ ⋮ ⋮ ⋮ ( − 1 ) y − 1 [ x C x ] ⋯ ( − 1 ) y − 1 [ x C 3 ] ( − 1 ) y − 1 [ x C 2 ] ( − 1 ) y − 1 [ x C 1 ] ( − 1 ) y − 1 [ x C 0 ] ]

Hex numbers:

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

Mod 6 of h_{1} ≡ 1; Mod 6 of h_{2} ≡ 1; ⋯ ; Mod 6 of h_{k} ≡ 1;

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

Theorem 2: ∑ k = 0 ∞ H 6 n ± k ≡ a 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 = ∏ i H i 6 = 1 .

We can say above as

R ( h 1 × h 2 × ⋯ × h k ) ÷ 6 ≡ ( 1 × 1 × ⋯ × 1 ) mod 6 = 1 .

Theorem 4: ∑ o H k mod ( 6 ) ≡ ∑ E H k ( mod 6 ) | k ∈ Z

Theorem 5: ∑ k = 0 ∞ H k = k 3 , where H_{k} is Hex series elements.

Matrices of Hex numbers

Let we see the relation between hex numbers in matrix

1) Let A = [ 1 h + 1 3 h + 1 6 h + 1 ] be a 2 × 2 matrix which elements are hex numbers (where h = 6) then | A | = | 1 h + 1 3 h + 1 6 h + 1 | = 6 h + 1 − 3 h 2 − 4 h + 1 = − 3 h 2 + 2 h

2) Let A = [ h + 1 3 h + 1 6 h + 1 10 h + 1 ] be a 2 × 2 matrix which elements are hex numbers (where h = 6) then | A | = | h + 1 3 h + 1 6 h + 1 10 h + 1 | = − 8 h 2 + 2 h

By above way we get, − 15 h 2 + 2 h ; − 24 h 2 + 2 h ; ⋯ ; − n ( n + 2 ) h 2 + 2 h

1) Let we construct a difference triangle about above determinants

a) Let A = [ 1 h + 1 3 h + 1 6 h + 1 10 h + 1 15 h + 1 21 h + 1 28 h + 1 36 h + 1 ] be a 3 × 3 matrix which elements are hex numbers then | A | = | 1 h + 1 3 h + 1 6 h + 1 10 h + 1 15 h + 1 21 h + 1 28 h + 1 36 h + 1 | = − 27 h 3

b) | A | = | h + 1 3 h + 1 6 h + 1 10 h + 1 15 h + 1 21 h + 1 28 h + 1 36 h + 1 45 h + 1 | = − 27 h 3

2) Let we construct a difference triangle about above determinants

a) Let A = [ 1 h + 1 3 h + 1 6 h + 1 10 h + 1 15 h + 1 21 h + 1 28 h + 1 36 h + 1 45 h + 1 55 h + 1 66 h + 1 78 h + 1 91 h + 1 105 h + 1 120 h + 1 ] be a 4 × 4 matrix which elements are hex numbers then | A | = | 1 h + 1 3 h + 1 6 h + 1 10 h + 1 15 h + 1 21 h + 1 28 h + 1 36 h + 1 45 h + 1 55 h + 1 66 h + 1 78 h + 1 91 h + 1 105 h + 1 120 h + 1 | = 0

b) | A | = | h + 1 3 h + 1 6 h + 1 10 h + 1 15 h + 1 21 h + 1 28 h + 1 36 h + 1 45 h + 1 55 h + 1 66 h + 1 78 h + 1 91 h + 1 105 h + 1 120 h + 1 136 h + 1 | = 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 2^{nd} difference.

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

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

Which means hex series elements are forming hexagonal only.

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.

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

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