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A New Generalization of Fibonacci Sequence & Extended Binet's Formula

Marcia Edson
  • Department of Mathematics & Statistics, Murray State University, Murray, KY 42071, USA. E-mail:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Omer Yayenie
  • Department of Mathematics & Statistics, Murray State University, Murray, KY 42071, USA. E-mail:
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2010-01-26 | DOI: https://doi.org/10.1515/INTEG.2009.051


Consider the Fibonacci sequence having initial conditions F 0 = 0, F 1 = 1 and recurrence relation Fn = F n–1 + F n–2 (n ≥ 2). The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this article, we study a new generalization {qn }, with initial conditions q 0 = 0 and q 1 = 1 which is generated by the recurrence relation qn = aq n–1 + q n–2 (when n is even) or qn = bq n–1 + q n–2 (when n is odd), where a and b are nonzero real numbers. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of {qn } with a = b = 1. Pell's sequence is {qn } with a = b = 2 and the k-Fibonacci sequence is {qn } with a = b = k. We produce an extended Binet's formula for the sequence {qn } and, thereby, identities such as Cassini's, Catalan's, d'Ocagne's, etc.

Keywords.: Fibonacci sequence; Generalized Fibonacci sequence; linear recurrence; characteristic sequence; generating function

About the article

Received: 2008-07-21

Revised: 2009-05-08

Accepted: 2009-08-08

Published Online: 2010-01-26

Published in Print: 2009-12-01

Citation Information: Integers, Volume 9, Issue 6, Pages 639–654, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/INTEG.2009.051.

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