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On Generalized Fibonacci Numbers

Yıl 2020, , 186 - 202, 22.12.2020

Fidel Oduol Isaac Owino Okoth

https://doi.org/10.33434/cams.771023
Cited By: 1

Öz

Fibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of $r$-sum Fibonacci numbers. The recurrence relation is maintained but initial conditions are varied. Among results obtained are Binet's formula, generating function, explicit sum formula, sum of first $n$ terms, sum of first $n$ terms with even indices, sum of first $n$ terms with odd indices, alternating sum of $n$ terms of $r-$sum Fibonacci sequence, Honsberger's identity, determinant identities and a generalized identity from which Cassini's identity, Catalan's identity and d'Ocagne's identity follow immediately.

Anahtar Kelimeler

Binet's formula Fibonacci sequence generating function r-shifted Fibonacci sequence

Destekleyen Kurum

Maseno University

Kaynakça

  • [1] S. Falcon, A. Plaza, On the Fibonacci K-numbers, Chaos Solution Fractals, 32(5) (2007), 1615–1624.
  • [2] Y.K Gupta, M. Singh, O. Sikhwal, Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences, Turkish J. Anal. Number Theory, 2(6) (2014), 233–238.
  • [3] A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 68(1961), 455–459.
  • [4] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fib. Quart, 3(3) (1965),161–176.
  • [5] D. Kalma, R. Mena, The Fibonacci Numbers-Exposed, Math. Mag., 2 (2002).
  • [6] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wisley-Interscience Publications, New York, 2011.
  • [7] Y.K. Panwar, M. Singh, Certain properties of generalized Fibonacci sequence, Turkish J. Anal. Number Theory, 2(1) (2014), 6–8.
  • [8] G.P.S Rathore, O. Sikhwal, R. Choudhary, Generalized Fibonacci-like sequence and some identities, SCIREA J. Math., 1(1)(2016), 107–118.
  • [9] O. Sikhwal, Y. Vyas, Generalized Fibonacci-type sequence and its Properties, Int. J. Sci. Res., 5(12) (2016), 2043–2047.
  • [10] B. Singh, S. Bhatnagar, Fibonacci-like sequence and its properties, Int. J. Contemp. Math. Sci., 5(18) (2010), 859–868.
  • [11] B. Singh, S. Bhatnagar, O. Sikhwal, Fibonacci-like sequence, Int. J. Adv. Math. Sci., 1(3)(2013), 145–151.
  • [12] M. Singh, Y. Gupta, O. Sikhwal, Identities of generalized Fibonacci-like sequence, Turkish J. Anal. Number Theory, 2(5) (2014), 170–175.
  • [13] B. Singh, O. Sikhwal, Y. K Gupta, Generalized Fibonacci-Lucas sequence, Turkish J. Anal. Number Theory, 2(6)(2014), 193–197.
  • [14] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences (OEIS), Available at http://oeis.org.
  • [15] A. Wani, G. P. S. Rathore, K. Sisodiya, On the properties of Fibonacci-Like sequence, Int. J. Math. Trends Tech., 29(2) (2016), 80–86.

Yıl 2020, , 186 - 202, 22.12.2020

Fidel Oduol Isaac Owino Okoth

https://doi.org/10.33434/cams.771023
Cited By: 1

Öz

Kaynakça

  • [1] S. Falcon, A. Plaza, On the Fibonacci K-numbers, Chaos Solution Fractals, 32(5) (2007), 1615–1624.
  • [2] Y.K Gupta, M. Singh, O. Sikhwal, Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences, Turkish J. Anal. Number Theory, 2(6) (2014), 233–238.
  • [3] A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 68(1961), 455–459.
  • [4] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fib. Quart, 3(3) (1965),161–176.
  • [5] D. Kalma, R. Mena, The Fibonacci Numbers-Exposed, Math. Mag., 2 (2002).
  • [6] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wisley-Interscience Publications, New York, 2011.
  • [7] Y.K. Panwar, M. Singh, Certain properties of generalized Fibonacci sequence, Turkish J. Anal. Number Theory, 2(1) (2014), 6–8.
  • [8] G.P.S Rathore, O. Sikhwal, R. Choudhary, Generalized Fibonacci-like sequence and some identities, SCIREA J. Math., 1(1)(2016), 107–118.
  • [9] O. Sikhwal, Y. Vyas, Generalized Fibonacci-type sequence and its Properties, Int. J. Sci. Res., 5(12) (2016), 2043–2047.
  • [10] B. Singh, S. Bhatnagar, Fibonacci-like sequence and its properties, Int. J. Contemp. Math. Sci., 5(18) (2010), 859–868.
  • [11] B. Singh, S. Bhatnagar, O. Sikhwal, Fibonacci-like sequence, Int. J. Adv. Math. Sci., 1(3)(2013), 145–151.
  • [12] M. Singh, Y. Gupta, O. Sikhwal, Identities of generalized Fibonacci-like sequence, Turkish J. Anal. Number Theory, 2(5) (2014), 170–175.
  • [13] B. Singh, O. Sikhwal, Y. K Gupta, Generalized Fibonacci-Lucas sequence, Turkish J. Anal. Number Theory, 2(6)(2014), 193–197.
  • [14] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences (OEIS), Available at http://oeis.org.
  • [15] A. Wani, G. P. S. Rathore, K. Sisodiya, On the properties of Fibonacci-Like sequence, Int. J. Math. Trends Tech., 29(2) (2016), 80–86.
Toplam 15 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Fidel Oduol 0000-0002-1228-6339

Isaac Owino Okoth

Yayımlanma Tarihi 22 Aralık 2020
Gönderilme Tarihi 17 Temmuz 2020
Kabul Tarihi 29 Eylül 2020
Yayımlandığı Sayı Yıl 2020

Kaynak Göster

APA Oduol, F., & Okoth, I. O. (2020). On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences, 3(4), 186-202. https://doi.org/10.33434/cams.771023
AMA Oduol F, Okoth IO. On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences. Aralık 2020;3(4):186-202. doi:10.33434/cams.771023
Chicago Oduol, Fidel, ve Isaac Owino Okoth. “On Generalized Fibonacci Numbers”. Communications in Advanced Mathematical Sciences 3, sy. 4 (Aralık 2020): 186-202. https://doi.org/10.33434/cams.771023.
EndNote Oduol F, Okoth IO (01 Aralık 2020) On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences 3 4 186–202.
IEEE F. Oduol ve I. O. Okoth, “On Generalized Fibonacci Numbers”, Communications in Advanced Mathematical Sciences, c. 3, sy. 4, ss. 186–202, 2020, doi: 10.33434/cams.771023.
ISNAD Oduol, Fidel - Okoth, Isaac Owino. “On Generalized Fibonacci Numbers”. Communications in Advanced Mathematical Sciences 3/4 (Aralık 2020), 186-202. https://doi.org/10.33434/cams.771023.
JAMA Oduol F, Okoth IO. On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences. 2020;3:186–202.
MLA Oduol, Fidel ve Isaac Owino Okoth. “On Generalized Fibonacci Numbers”. Communications in Advanced Mathematical Sciences, c. 3, sy. 4, 2020, ss. 186-02, doi:10.33434/cams.771023.
Vancouver Oduol F, Okoth IO. On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences. 2020;3(4):186-202.

Cited By

Tessarine Number Sequences and Quantum Calculus Approach
Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi
https://doi.org/10.53433/yyufbed.1595620

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