On Generalized Fibonacci Numbers

Year 2020, , 186 - 202, 22.12.2020

Fidel Oduol Isaac Owino Okoth

https://doi.org/10.33434/cams.771023

Cited By: 1

Abstract

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.

Keywords

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

Supporting Institution

Maseno University

References

• [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.