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

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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 BY 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 © 2014 British Journals ISSN 2047-3745 British Journal of Science 63 September 2014, Vol. 11 (2) 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 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 © 2014 British Journals ISSN 2047-3745 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). © 2014 British Journals ISSN 2047-3745 British Journal of Science 65 September 2014, Vol. 11 (2) 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 © 2014 British Journals ISSN 2047-3745 British Journal of Science 66 September 2014, Vol. 11 (2) 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 © 2014 British Journals ISSN 2047-3745 British Journal of Science 67 September 2014, Vol. 11 (2) 15 610 Every third Fibonacci number is divisible by two, or F3.
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