The Existence of Fibonacci Numbers in the Algorithmic Generator for Combinatoric Pascal Triangle

British Journal of Science 62 September 2014, Vol. 11 (2)

THE EXISTENCE OF FIBONACCI NUMBERS IN THE ALGORITHMIC GENERATOR FOR COMBINATORIC PASCAL TRIANGLE

Amannah, Constance Izuchukwu

[email protected]; +234 8037720614

Department of Computer Science, Faculty of Natural and Applied Sciences, Ignatius Ajuru University of Education, P.M.B. 5047, Port Harcourt, Rivers State, Nigeria.

Nanwin, Nuka Domaka Department of Computer Science, Faculty of Natural and Applied Sciences, Ignatius Ajuru University of Education, P.M.B. 5047, Port Harcourt, Rivers State, Nigeria

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ABSTRACT

The discoveries of Leonard of Pisa, better known as Fibonacci, are revolutionary contributions to the mathematical world. His best-known work is the Fibonacci sequence, in which each new number is the sum of the two numbers preceding it. When various operations and manipulations are performed on the numbers of this sequence, beautiful and incredible patterns begin to emerge. The numbers from this sequence are manifested throughout nature in the forms and designs of many plants and animals and have also been reproduced in various manners in art, architecture, and music. This work simulated the Pascal triangle generator to produce the Fibonacci numbers or sequence. The Fibonacci numbers are generated by simply taken the sums of the "shallow" diagonals (shown in red) of Pascal's triangle. The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle. This Pascal triangle generator is a combinatoric algorithm that outlines the steps necessary for generating the elements and their positions in the rows of a Pascal triangle. The Pascal triangle generator is symbolized i with E j. The is denote the element of a row while the js represent the respective positions of the elements. The generated Fibonacci sequence from i the E j model can be used in the following way; in the computational run- time analysis of Euclid's algorithm to determine the greatest common divisor of two integers- the worst case input for this algorithm is a pair of consecutive Fibonacci numbers; as pseudorandom number generators; The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most. This work succeeded in simulating the Pascal triangle to produce 20 Fibonacci numbers namely; 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765.

KEY WORDS: Algorithm, Generator, Triangle, Fibonacci Numbers, Sequence, Shallow Diagonal

INTRODUCTION

British Journal of Science 64 September 2014, Vol. 11 (2) nature in the forms and designs of many plants and animals and have also been reproduced in various manners in art, architecture, and music. The mathematician Leonardo of Pisa, better known as Fibonacci, had a significant impact on mathematics. His contributions to mathematics have intrigued and inspired people through the centuries to delve more deeply into the mathematical world. He is best known for the sequence of numbers bearing his name.

Leonardo Pisano Bigollo (c. 1170 – c. 1250) [Wikipedia) – known as Fibonacci, and also Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci – was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages, Howard (1990). Fibonacci is best known to the modern world for (Encyclopædia Britannica), the spreading of the Hindu–Arabic numeral system in Europe, primarily through his composition in 1202 of Liber Abaci (Book of Calculation), and for a number sequence named the Fibonacci numbers after him, which he did not discover but used as an example in the Liber Abaci, (Parmanand, 1986).

The Fibonacci sequence is named after Leonardo Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, Goonatilake (1998), although the sequence had been described earlier in Indian mathematics. (Knuth 2006; Singh 1985; Douady and Couder 1996). By modern convention, the sequence begins either with F0 = 0 or with F1 = 1. The Liber Abaci began the sequence with F1 = 1, without an initial 0.

Fibonacci numbers are closely related to Lucas numbers in that they are a complementary pair of Lucas sequences. They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5,.... Applications include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, Jones and Wilson (2006) such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineapple, Brousseau (1969), the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. Knuth (2008). The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody,(Singh 1985; Knuth 1968). In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and 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, (Knuth 2006).

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Liber Abaci also posed, and solved, a problem involving the growth of a population of rabbits based on idealized assumptions. The solution, generation by generation, was a sequence of numbers later known as Fibonacci numbers. The number sequence was known to Indian mathematicians as early as the 6th century Donald (2006) and Rachel (2008), but it was Fibonacci's Liber Abaci that introduced it to the West.

In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0, 1, 1, 2, as modern mathematicians do but with 1, 1, 2, etc. He carried the calculation up to the thirteenth place (fourteenth in modern counting), that is 233, though another manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.

The problem, dealing with the regeneration of rabbits, calculated the number of rabbits after a year if there is only one pair the first month. The problem states that it takes one month for a rabbit pair to mature, and the pair will then produce one pair of rabbits each month following. Fibonacci’s solution stated that in the first month there would be only one pair; the second month there would be one adult pair and one baby pair; the third month there would be two adult pairs and one baby pair; and so forth (Posamentier and Lehmann, 2007). When the total number of rabbits for each month is listed, one after the other, it generates the sequence of numbers for which Fibonacci is most famous:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…

This string of numbers is known as the Fibonacci sequence, and each successive term is found by adding the two preceding terms together. The Fibonacci sequence is the oldest known recursive sequence, which is a sequence where each successive term can only be found through performing operations on previous terms. Interestingly, Fibonacci does not comment on the recursive nature of this sequence. The relationship between the terms was not identified in publication until four hundred years later. At the time of the publication of Liber Abaci, no special notice was taken of these numbers. It was not until the mid- 1800s that mathematicians began to be intrigued by what would later be known as the Fibonacci numbers (Posamentier and Lehmann, 2007). A closer inspection of the numbers making up the Fibonacci sequence brings to light all sorts of fascinating patterns and mathematical properties. Fibonacci himself makes no mention of these patterns in his book, but the following patterns are a few that have been brought to light over years of examination of the numbers in the sequence. Any two consecutive Fibonacci numbers are relatively prime, having no factors in common with each other (Garland, 1987). For example: 5, 8,13,21,34

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5 = 1 · 5; 8 = 2 · 2 · 2; 13 = 1 · 13; 21 = 3 · 7; 34 = 2 · 17

Summing together any ten consecutive Fibonacci numbers will always result in a number which is divisible by eleven (Posamentier and Lehmann, 2007).

1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143

143/11 = 13

89 + 144 + 233 + 377 + 610 + 987 + 1,597 + 2,584 + 4,181 + 6,675 = 17,567

17,567/11 = 1,597

Following tradition, Fn will be used to represent the n-th Fibonacci number in the sequence. n Fn

1 1

2 1

3 2

4 3

5 5

6 8

7 13

8 21

9 34

10 55

11 89

12 144

13 233

14 377

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15 610

Every third Fibonacci number is divisible by two, or F3. Every fourth Fibonacci number is divisible by three, or F4. Every fifth Fibonacci number is divisible by five, or F5. Every sixth Fibonacci number is divisible by eight, or F6, and the pattern continues. In general, every nth Fibonacci number is divisible by the nth number in the Fibonacci sequence, or FMN is divisible by FN (Garland, 1987). Fibonacci numbers in composite-number positions are always composite numbers, with the exception of the fourth Fibonacci number. In other words if n is not a prime, the nth Fibonacci number will not be a prime (Posamentier and Lehmann, 2007)

35).

F6 = 8

F9 = 34

F16 = 987

The reciprocal of the eleventh Fibonacci number, 89, can be found by adding the Fibonacci sequence in such a fashion that each Fibonacci number contributes one digit to the repeating decimal of the reciprocal, 1/89 (Garland, 1987).

0.0112358

144

233

377

610

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987

1/89 = 0.01123595505617787

Multiplying any Fibonacci number by two and subtracting the next number in the sequence will result in the answer being the number two places before the original (Garland, 1987).

…3,5,8,13,21,34,55,89,144,233…

2 · F6 –F7 = (2 · 8) - 13 = 16 - 13 = 3 = F4

2 · F11- F12= (2 · 89) - 144 = 178 - 144 = 34 – F9

2 · Fn – Fn+1= Fn-2

Summing consecutive odd-positioned Fibonacci numbers, starting with the first odd-positioned number, F1, will result in a number that is the next Fibonacci number in the sequence after the last term in the sum (Posamentier and Lehmann, 2007).

F1 + F3 = 1 + 2 = 3 = F4

F1 + F3 + F5 = 1 + 2 + 5 = 8 = F6

F1+ F3+ F5 +F7 = 1 + 2 + 5 + 13 = 21 = F8

A similar pattern emerges when summing consecutive, even-positioned Fibonacci numbers beginning with F2, only this time, the result is a number that is one less than the Fibonacci number following the last even number in the sum (Posamentier and Lehmann, 2007).

F2 + F4 = 1 + 3 = 4 = F5 - 1

F2 + F4 + F6 = 1 + 3 + 8 = 12 = F7 - 1

F2 + F4 + F6 + F8 = 1+ 3 + 8 + 21 = 33 = F9 - 1

The product of any Fibonacci number multiplied by the number two places after it will be one more or one less than the square of the Fibonacci number between the two. When the number to be squared is an even-positioned Fibonacci number, one is added, and when it is odd-positioned, one is subtracted (Posamentier & Lehmann, 2007).

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…3,5,8,13,21,34,55,89…

2 2 F4 · F6 = 3 · 8 = 24 and F 5= 5 = 25

2 2 F9 · F11= 34 · 89 = 3,026 and F 10= 55 = 3,025

Pascal triangle on the other hand, measures trend of proportion in probability, where it is used to find combinations. There are a number of principles that are employed in solving combiantoric problems. These mathematical principles include the Pigeon Hole Principle, the Stocks in Drawer Principle, the Counting

Principle, the Multiplication Principle, the Permutation Principle, the

Combination Principle and the Scheduling Principle.

Interesting, as these principles have proved to be, in their various numerical results, the demands for a more technical and mathematically admissible algorithm for an efficient generation of a scope free, nth-term Pascal triangle remains in the dream of scientific computing. A synchronizing study of the aforementioned combinatoric principles, undisputably translates this scientific expectation to computing reality. The Pascal Generating algorithm in paper provides a computing and mathematical model for achieving an nth term Pascal

Triangle. The Lemma, Pascal rule was introduced into the West by Blaise

Pascal, Riordan (1958).

The goal is to provide a unified treatment of analysis in combinatorics. Verlag’s

(2002) analysis as the core of the theory focuses on rational and meromorphic

British Journal of Science 70 September 2014, Vol. 11 (2) functions as well as two of their explanations given by verlag on fundamentals of singularity analysis and combinatorial consequences, Stanley (1997; 1999).

1.1 Applications of Fibonacci Numbers

 The Fibonacci numbers are important in the computational run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers: the worst case input for this algorithm is a pair of consecutive Fibonacci numbers, Knuth (1997).  Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem.  The Fibonacci numbers are also an example of a complete sequence. This means that every positive integer can be written as a sum of Fibonacci numbers, where any one number is used once at most.  Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. The Zeckendorf representation of a number can be used to derive its Fibonacci coding.  Fibonacci numbers are used by some pseudorandom number generators.  Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ. A tape-drive implementation of the polyphase merge sort was described in The Art of Computer Programming.  Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.  The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.  A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers.

2 METHODOLOGY

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The research design adopted in this work is simulation research. It simulated the Pascal triangle generator to demonstrate the possibility of the Fibonacci numbers. In the Pascal triangle we could see the incidence of the Fibonacci numbers as much as we could see the angle of incidence in a prison experiment in Physics. The direction of incidence of the Fibonacci numbers is along the shallow diagonals of the Pascal triangle.

3 THE ALGORITMIC GENERATOR FOR THE COMBINATORIC PASCAL TRIANGLE

3.1 The Lemma of Blaise Pascal

The formula for counting the number of ways to select m elements from a set

of n total elements can be characterized by this shorthand notation.

 n  n!    - - - (1)  m  m!(n  m)!

With this prelude, we can proceed to look at Pascal’s Triangle. The triangle builds as follows.

(n=0) 1

(n=1) 1 1

(n=2) 1 2 1

(n=3) 1 3 3 1

In each case, the number in the next row is equal to the sum of the two numbers directly above it.

Let us deduce certain principle that will enable us in preparing our model. Let each item in a Pascal triangle be called element, denoted, by E. Let also all elements in Row 1 be denoted by E1 and for elements in row 2, E3 for Row3,

British Journal of Science 72 September 2014, Vol. 11 (2) and Ei for Rowi for all i = 1 to nth row. The next step will be to represent the position of each element given any nth number of rows, let Ej denote the elements positions in the rows for all j = l to nth elements. E1 = First position element, E2 = second position element, E 3 = third position element of the row in question. coronary 1:

Because the Pascal triangle is symmetric, E1 (clockwise) = E1 (anti-clockwise).

The centre of the row is the negotiation limit of the triangle. At this point, we decide where to move, either to the left or to the right. All Ej’s from the NL to the left = All Ej’s from the NL to the right.

Synthesizing the algorithm we have

ic E jc = elements in rows is in positions js

Vi = 1, 2, 3 … nth element and

Vj = 1, 2, 3 . . . nth position

c = NL = Negotiating limit

Let’s see how this maps into the equation (a + b) n

(a + b)0 = (1) 1

(a + b)1 = (1) a + (1) b

(a+b)2 = (1)a2 (2) ab + (1) b2

(a+b)3 = (1)a3 + (3)a2b + (3)ab2 + (1)b3

(a+b)4 = (1)a4 + (4)a3b+ (6)a2b2 + (4) ab3 + (1)b4

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If we remember that;

0! = 1, we get the following;

For n = 0, the single element is equal to;

 0  0!    (0  0)! = 1  0  0!

 1  1  3   3  3 For n = 1,    1,    1 For n = 2,    1,    3,    1  0  1  0   1   3

This is useful because it allows us to pick any exponent n and generate the list of coefficients that makes up its values.

3.2 Proof of Pascal Rule

 n   n 1  n 1          - - - (2)  m   m   m 1

 n 1 (n 1)! (n 1)!    - - - (3)  m  m![(n 1)  m]! m!(n  m 1)!

 n 1  (n 1)!    - - - (4)  m 1 (m 1)! [(n 1)  (m 1)]!

(n 1)! (n 1)! =  | (m 1)! (n 1 m 1)! (m 1)!(n  m)!

The trick here is to add both of fractional values.

M! = (m) (m-1)! - - - (5)

(n-m)! = (n-m) (n-m-1)! - - - (6)

Applying (5) and (6) to (2) and (3) we have

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 n 1  n 1  (n  m)(n 1)! (m)(n 1)!         m   m 1 m!(n  m)! m!(n  m)!

[(n  m)  m](n 1)! n! =  - - - (7) m!(n  m)! m1(n  m)!

The resolution shows that equations (1) and (7) are equal. n  n 1  n 1  n!        - - - (8) m  m   m 1 m!(n  m)!

Row 1: = E1

1 1 Row 2: = E 1, E1

2 2 , 1 1 2 2 Row 3: = E 1 + E 2 E 1 or E 1, E 1 + E 2

1 3 3 Row 4: = E 1, E 1, + E 2

1 4 4 4 Row 5: = E 1, E 1 + E 2, E 2 + E+43

1 5 5 5 5 Row 6: = E 1, E 1 + E 2, E 2 + E 3

1 6 6 6 6 6 6 Row 7: = E 1, E 1 + E 2, E 2, E 3, E 3+ E 4

1 7 7 7 7 7 7 7 7 Row 8: = E 1, E 1, + E 2, E 2, E 3, E 3+ E 4, E 4 + E 5

1 8 8 8 8 8 8 8 8 8 Row 9: = E 1, E 1 + E 2, E 2 + E 3, E 3 + E 4, E 4, E 5, E 5

8 E 6

1 9 9 9 9 9 9 9 9 Row 10: = E 1, E 1 + E 2, E 2+ E 3, E 3 + E 4, E 4 + E 5,

9 9 9 9 E 5 + E 6, E 6 + E 7

1 10 10 10 10 10 10 10 10 10 10 Row 11: = E 1, E 1 + E 2, E 2 + E 3, E 3 + E 4, E 4 + E 5, E 5 + E 6,

10 10 10 10 E 6 + E 7, E 7 + E 8

1 11 11 11 11 11 11 11 11 10 11 Row 12: = E 1, E 1 + E 2, E 2 +E 3, E 3 +E 4, E 4+E 5, E 5 + E 6, 11 11 11 11 11 11 E 6 + E 7, E 7 + E 8,E 8+ E 9

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1 12 12 12 12 12 12 12 12 Row 13: = E 1, E 1+ E 2, E 2 +E 3, E 3 +E 4, E 4+E 5, 12 12 E 5 + E 6, 1 13 13 13 13 13 13 13 Row 14: = E 1, E 1+ E 2, E 2 +E 3, E 3 +E 4, E 4 13 13 13 13 13 13 13 13 +E 5,E 5 + E 6, E 6 + E 7, E 7 + E 8,E 8 + 13 13 13 13 13 E 9, E 9+ E 10, E 10 + E 11 1 14 14 14 14 14 14 14 14 14 14 Row 15: = E 1, E 1+ E 2, E 2 +E 3, E 3 +E 4, E 4 +E 5, E 5 + E 6, 14 14 14 12 12 12 12 12 12 12 12 E 6 + E 7, E 7 +E 6 + E 7, E 7 + E 8,E 8+ E 9, E 9 +E 10 1 15 15 15 15 15 15 15 15 15 15 Row 16: = E 1, E 1 + E 2, E 2 +E 3, E 3 +E 4, E 4 +E 5, E 5 + E 6,

15 15 15 15 15 15 15 15 15 E 6 + E 7, E 7 + E 8, E 8 + E 9, E 9+ E 10, E 10 +

15 15 15 15 15 E 11, E 11 + E 12, E 12 +E 13.

1 i-1 i-1 i-1 i-1 i-1 Rowij : = E 1, E 1 + E E1+ 1, E E1+1 +E E1+2 + . . . E C

Fig.1: The Pascal Triangle Algorithmic Generator

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3001 1365 455 105 15 1

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1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1

1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1

1 18 153 816 3060 8568 18566 31724 43758 48620 43758 31724 31724 18566 8568 3060 816 153 18 1

1 19 171 969 3876 11628 27134 40290 75482 82378 92378 75482 40290 27134 11628 3876 969 171 19 1

1 20 162 1140 4845 15504 38762 67424 115772 157860 175756 177860 115772 67424 38762 15504 4845 1140 162 20 1 Fig. 2: Results of the Pascal Generator

The combinatoric algorithm represents half-symmetry, ½C of the Pascal i triangle. A complete symmetry is the collection of the E ; of both right and left, from NL or C, where C is the centre. It is of little or no essence to repeat ½C from left of right or vice versa. So we terminate our algorithm at half symmetry ½C.

1. The Fibonacci Model

By definition, the first two numbers in the Fibonacci sequence are 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.

or (often, in modern usage):

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

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4.1 List of Fibonacci Numbers

The first 21 Fibonacci numbers Fn for n = 0, 1, 2, ..., 20 are: Bona (2000)

F F F F F F F F F F F1 F1 F12 F13 F14 F15 F16 F17 F18 F19 F20 0 1 2 3 4 5 6 7 8 9 0 1 1 2 3 14 23 37 61 98 159 258 418 676 0 1 1 2 3 5 8 55 89 3 1 4 4 3 7 0 7 7 4 1 5

The sequence can also be extended to negative index n using the re-arranged recurrence relation

which yields the sequence of "negafibonacci" numbers Lemmeryer (2000) satisfying

Thus the bidirectional sequence is

F−8 F−7 F−6 F−5 F−4 F−3 F−2 F−1 F0 F1 F2 F3 F4 F5 F6 F7 F8 −21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13 21

4.2 Existence of Fibonacci Numbers in Pascal Triangle

Fig. 3: Generating Fibonacci Numbers from Pascal Triangle

Figure 3 above contains Fibonacci numbers for n+10. The Fibonacci numbers, fn =1,1,2,3,5,8,13,21,34,55,89

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Fi=1, f2=1, f3=2, f4=3, f5=5, f6=8, f7=13, f8=21, f9=34, f10=55

0 0 1 1 1 1 2 1+1 3 1+2 5 2+3 8 3+5 13 5+8 21 8+13 34 13+21 55 21+34 89 34+55 144 55+89 233 89+144 377 144+233 610 233+377 987 377+610 1597 610+987 2584 987+1597 4181 1597+2584 6765 2584+4181 10946 4181+6765 17711 6765+10946 28657 10946+17711 46368 17711+28657 75025 28657+46368

The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of Pascal's triangle. The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle. Lucas (1891).

These numbers also give the solution to certain enumerative problems. Pisano(2002). The most common of such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n: there are Fn+1 ways to do this. For example F6 = 8 counts the eight compositions:

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1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 6−1 = 5.

This configuration has many interesting and important properties:

 Notice the left-right symmetry - it is its own mirror image.  Notice that in each row, the second number counts the row.  Notice that in each row, the 2nd + the 3rd counts the number of numbers above that line.

There are endless variations on this theme.

Next, notice what happens when we add up the numbers in each row - we get our doubling sequence.

Now for visual convenience we draw the triangle left-justified. Add up the numbers on the various diagonals...

British Journal of Science 80 September 2014, Vol. 11 (2) and we get 1, 1, 2, 3, 5, 8, 13, . . . the Fibonacci sequence!

4.3 Applications of Fibonacci Numbers

 Fibonacci numbers are used by some pseudorandom number generators.  Fibonacci numbers are used in a polyphase version of the merge sort algorithm in which an unsorted list is divided into two lists whose lengths correspond to sequential Fibonacci numbers – by dividing the list so that the two parts have lengths in the approximate proportion φ.  Fibonacci numbers arise in the analysis of the Fibonacci heap data structure.  The Fibonacci cube is an undirected graph with a Fibonacci number of nodes that has been proposed as a network topology for parallel computing.  A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers, Knuth (1997).

4.4 Fibonacci primes

A Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, …

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.

5. RESULTS AND DISCUSSION

Figure 4 below contains the first 20 Fibonacci numbers from the generated Pascal triangle. This is the result of simple summation of the shallow diagonal the Pascal triangle.

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Fig. 4: Resultant Fibonacci Numbers from the Pascal Triangle

5. CONCLUSION

This unique and fascinating string of numbers possesses all sorts of intriguing properties, which can be discovered by applying various mathematical procedures to the numbers in the sequence. Fibonacci numbers are present throughout the world in which we live, and the patterns which can be formed from them both astonish and perplex the mind. The Fibonacci numbers are beautiful to study in and of themselves, but there is a higher beauty to them as

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These numbers highlight the incredible order and mathematical complexity of the world we live in, which all points to the Creator. Such intricacies could not have evolved by mere chance, but are the work of a God of order, who created all things.

Fibonacci numbers can be applied to an elementary example of geometric growth - asexual reproduction, like that of the amoeba. Each organism splits into two after an interval of maturation time characteristic of the species. This interval varies randomly but within a certain range according to external conditions, like temperature, availability of nutrients and so on. We can imagine a simplified model where, under perfect conditions, all amoebae split after the same time period of growth. So, one amoeba becomes two, two become 4, then 8, 16, 32, and so on.

Fig. 5: Doubling Sequence in asexual Reproduction in Amoeba

We get a doubling sequence. Notice the recursive formula:

 An =2An

This of course leads to exponential growth, one characteristic pattern of population growth.

REFERENECS

Ball, K. M. (2003). Fibonacci's Rabbits Revisited, Strange Curves, Counting Rabbits, and Other Mathematical Explorations, Princeton, NJ: Princeton University Press.

Beck, M. and Geoghegan, R. (2010). The Art of Proof: Basic Training for Deeper Mathematics, New York: Springer.

Bóna, M. (2011). A Walk Through Combinatorics (3rd ed.), New Jersey: World Scientific.

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Brooks, C. (1990). An Introduction to the History of Mathematics.: (6th ed.),

Brousseau, A (1969), "Fibonacci Statistics in Conifers", Fibonacci Quarterly (7): 525–32

Donald, K. (2006). The Art of Computer Programming: Generating All Trees – History of Combinatorial Generation; Volume 4. Addison-Wesley. p. 50

Douady, S. and Couder, Y. (1996). Phyllotaxis as a Dynamical Self Organizing Process (PDF), Journal of Theoretical Biology 178 (178): 255–74.

Garland, T. H. (1987). Fascinating Fibonaccis: Mystery and magic in numbers. Palo Alto, CA: Dale Seymour.

Goonatilake, S, (1998), Toward a Global Science, Indiana University Press, p. 126.

Howard, E. (1990). An Introduction to the History of Mathematics. Brooks Cole: (6th ed.), p 261.

Jones, J. and Wilson, W. (2006). Science, an Incomplete Education, Ballantine Books, p. 544.

Knuth, D. (2006). The Art of Computer Programming, 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50.

Knuth, D. E. (1997). The Art of Computer Programming,: Fundamental Algorithms (3rd ed.), Addison–Wesley, p. 343

Knuth, D. E. (1997). The Art of Computer Programming: Fundamental A lgorithms (3rd ed.), Addison–Wesley, p. 343,

Knuth, Donald (2008). Negafibonacci Numbers and the Hyperbolic Plane, Annual meeting, The Fairmont Hotel, San Jose, CA: The Mathematical Association of America.

Lemmermeyer, F. (2000). Reciprocity Laws, New York: Springer.

Leonardo Pisano – page 3: Contributions to number theory. Encyclopædia Britannica Online, 2006. Retrieved January 14 2014.

Lucas, É. (1891). Théorie des nombres (in French) 1, Gauthier-Villars.

p 261

British Journal of Science 84 September 2014, Vol. 11 (2)

Parmanand, S. (1986). Acharya Hemachandra and the (so called) Fibonacci Numbers. Math. Ed. Siwan , 20(1):28–30.

Pisano, L. (2002). Fibonacci's Liber Abaci: A Translation into Modern English of the Book of Calculation (hardback), Sources and Studies in the History of Mathematics and Physical Sciences, Sigler, Laurence E, trans, Springer,

Posamentier, A. S., & Lehmann, I. (2012). The glorious golden ratio. Amherst, NY: Prometheus Books. Rachel, W. H. (2008). Math for poets and drummers. Math Horizons (15) 10- 11.

Singh, P. (1985). The So-called Fibonacci numbers in ancient and medieval India, Historia Mathematica 12 (3): 229–44.

Susantha, G. (1998). Toward a Global Science. Indiana University Press. p. 126. The Fibonacci Series – Biographies – Leonardo Fibonacci (ca.1175 – ca.1240)". Library.thinkquest.org. Retrieved January 14 2014.