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Pascal Üçgeni, Kombinasyon ve Tümevarım Kullanarak Fibonacci Dizisinin N. Elemanını Bulma

Year 2017, Volume: 4 Issue: 3, 429 - 435, 30.09.2017
https://doi.org/10.31202/ecjse.317750

Abstract

Bilindiği
üzere Fibonacci dizisi bilişim teknolojileri dâhil birçok mühendislik alanında
kullanılmaktadır. Fibonacci dizisinin n.
elemanını bulabilmek için (n-1). ve (n-2). elemanlarının da hesaplanması
gerekir. Bu işlem bilinmeyen her bir elemanın hesaplanması işlemi özyinelemeli
olarak 1. ve 2. elemana kadar gider. Bu çalışmada Paskal üçgeninden faydalanılarak
Fibonacci dizisinin n. elemanını doğrudan
bulabilen bir formül önerilmiştir. Bilindiği üzere Paskal üçgenine sol alttan sağ
yukarı doğru diagonal düzlemdeki tüm elemanlar toplandığında Fibonacci dizisinin
elemanları sırayla hesaplanabilmektedir. Bu düzlemde gizli olarak bulunan örüntü,
matematikteki Kombinasyon, Tümevarım ve Fonksiyon konuları ile modellenerek
yeni bir formül haline dönüştürülmüştür. Fibonacci serisindeki elemanları bulmak
için özyinelemeli ve dinamik programlama yöntemleri ile yapılan hesaplamalara
göre daha az zaman ve alan karmaşıklığı ile benzer sonuçlar bulunmuştur.

References

  • [1] Hsu C.H., Hung-Son D., "The application of Fibonacci sequence and Taguchi method for investigating the design parameters on spiral micro-channel." Applied System Innovation (ICASI), 2016 International Conference on IEEE, 2016.
  • [2] Plofker K., Hannah J., "Mathematics in India." Aestimatio: Critical Reviews in the History of Science 7, 45-53, 2015.
  • [3] Goel, N.S., Richter N., Stochastic models in biology, Elsevier, USA, 2016.
  • [4] Brasch TV. Byström J., Lystad L.P., "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, 2012.
  • [5] Orozco-Henao, C., "Active distribution network fault location methodology: A minimum fault reactance and Fibonacci search approach.", International Journal of Electrical Power & Energy Systems 84, 232-241, 2017.
  • [6] Kaplan H., Tarjan R.E., Zwick U., "Fibonacci heaps revisited." arXiv preprint arXiv:1407.5750, 2014).
  • [7] Klavžar S., "Structure of Fibonacci cubes: a survey." Journal of Combinatorial Optimization 25(4):505-522, 2013.
  • [8] Stakhov A.P., Massingue V., Sluchenkova A., "Introduction into Fibonacci coding and cryptography." Osnova, Kharkov, 1999.
  • [9] Lengyel T., "A counting based proof of the generalized Zeckendorf's theorem." Fibonacci Quarterly 44.4, 324, 2006.
  • [10] Knuth, D.E., The Art of Computer Programming, 1: Fundamental Algorithms (3rd ed.), Addison–Wesley, p. 343, ISBN 0-201-89683-4, 1997.
  • [11] DeBellis, R.S., Ronald M.S., Phil C.Y., "Pseudorandom number generator." U.S. Patent No. 6,044,388. 28 Mar. 2000.
  • [12] Cohn, M., Agile estimating and planning. Pearson Education, ISBN-13: 978-0131479418, 1st Edition, 2005.
  • [13] Niemann Thomas, Sorting and Searching Algorithms kitabı, Oregon, USA, 2010.
  • [14] Edson M, Yayenie O., "A New Generalization of Fibonacci Sequence & Extended Binet's Formula." Integers, 9(6, 639-654, 2009.

Calculating Nth Element of Fibonacci Sequence using Pascal Triangle, Combination and Sigma Symbol

Year 2017, Volume: 4 Issue: 3, 429 - 435, 30.09.2017
https://doi.org/10.31202/ecjse.317750

Abstract

As
is known, Fibonacci sequence is used in many engineering fields including
information technology. It is an obligation to calculate the (n-1)th and (n-2)th elements in the Fibonacci Sequence in order to find the (n)th element. These calculations
recursively go to the 1st and 2nd elements. In this
study, the Pascal triangle is used to determine the Fibonacci sequence. A
formula has been proposed that can find the required Fibonacci element
directly. It is known that the elements of the Fibonacci sequence can be
calculated sequentially when all the elements in the diagonal plane are
collected from left to right in the Pascal triangles. The hidden pattern  in this triangle is transformed into a new
formula by modeling with Combination, sigma symbol and Functions in
mathematics. To find the Fibonacci series, time and space complexity is reduced
to the minimum according to calculations made by recursive and dynamic
programming.

References

  • [1] Hsu C.H., Hung-Son D., "The application of Fibonacci sequence and Taguchi method for investigating the design parameters on spiral micro-channel." Applied System Innovation (ICASI), 2016 International Conference on IEEE, 2016.
  • [2] Plofker K., Hannah J., "Mathematics in India." Aestimatio: Critical Reviews in the History of Science 7, 45-53, 2015.
  • [3] Goel, N.S., Richter N., Stochastic models in biology, Elsevier, USA, 2016.
  • [4] Brasch TV. Byström J., Lystad L.P., "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, 2012.
  • [5] Orozco-Henao, C., "Active distribution network fault location methodology: A minimum fault reactance and Fibonacci search approach.", International Journal of Electrical Power & Energy Systems 84, 232-241, 2017.
  • [6] Kaplan H., Tarjan R.E., Zwick U., "Fibonacci heaps revisited." arXiv preprint arXiv:1407.5750, 2014).
  • [7] Klavžar S., "Structure of Fibonacci cubes: a survey." Journal of Combinatorial Optimization 25(4):505-522, 2013.
  • [8] Stakhov A.P., Massingue V., Sluchenkova A., "Introduction into Fibonacci coding and cryptography." Osnova, Kharkov, 1999.
  • [9] Lengyel T., "A counting based proof of the generalized Zeckendorf's theorem." Fibonacci Quarterly 44.4, 324, 2006.
  • [10] Knuth, D.E., The Art of Computer Programming, 1: Fundamental Algorithms (3rd ed.), Addison–Wesley, p. 343, ISBN 0-201-89683-4, 1997.
  • [11] DeBellis, R.S., Ronald M.S., Phil C.Y., "Pseudorandom number generator." U.S. Patent No. 6,044,388. 28 Mar. 2000.
  • [12] Cohn, M., Agile estimating and planning. Pearson Education, ISBN-13: 978-0131479418, 1st Edition, 2005.
  • [13] Niemann Thomas, Sorting and Searching Algorithms kitabı, Oregon, USA, 2010.
  • [14] Edson M, Yayenie O., "A New Generalization of Fibonacci Sequence & Extended Binet's Formula." Integers, 9(6, 639-654, 2009.
There are 14 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Makaleler
Authors

Faruk Bulut

Publication Date September 30, 2017
Submission Date May 31, 2017
Acceptance Date August 20, 2017
Published in Issue Year 2017 Volume: 4 Issue: 3

Cite

IEEE F. Bulut, “Pascal Üçgeni, Kombinasyon ve Tümevarım Kullanarak Fibonacci Dizisinin N. Elemanını Bulma”, El-Cezeri Journal of Science and Engineering, vol. 4, no. 3, pp. 429–435, 2017, doi: 10.31202/ecjse.317750.
Creative Commons License El-Cezeri is licensed to the public under a Creative Commons Attribution 4.0 license.
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