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On Generalized Fibonacci Numbers 1

Profile image of Jerico BacaniJerico Bacani

2018

Abstract

We provide a formula for the nth term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation x+x−k = 2. We then extend our results to k-generalized Horadam (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for k-periodic k-nary Fibonacci sequence can also be derived. Mathematics Subject Classification: 11B39, 11B50.

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References (8)

  1. Alp, M., Irmak, N., and Szalay, L., Two-periodic ternary recur- rences and their binet-formula, Acta Math. Univ. Comenianae, Vol. LXXXI, 2 (2012), pp. 227-232.
  2. Dresden, G. P. B., A Simplifed Binet Formula for k-Generalized Fi- bonacci Numbers, J. Integer Sequences, 19, (2013).
  3. Edson, M., Yayenie, O., A New Generalization of Fibonacci Sequence and Extended Binets Formula, Integers, 9 (# A48) (2009), pp. 639-654.
  4. Edson, M., Lewis, S., Yayenie, O., The k-periodic Fibonacci se- quence and an extended Binet's formula, Integers, 11 (# A32) (2011), pp. 639-652.
  5. Horadam, A. F., Basic properties of certain generalized sequence of numbers, Fibonacci Quarterly, 3 (1965), pp. 161-176.
  6. Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley, New York, 2001.
  7. Noe, Tony; Piezas, Tito III; and Weisstein, Eric W. "Fi- bonacci n-Step Number." From MathWorld -A Wolfram Web Resource. http://mathworld.wolfram.com/Fibonaccin-StepNumber.html
  8. Wu, Z., Zhang, H., On the reciprocal sums of higher-order sequences, Adv. Diff. Equ., 2013, 2013:189. Received: March 3, 2015

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