A New Generalization of Fibonacci Sequence & Extended Binet's Formula
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Marcia Edson
Abstract
Consider the Fibonacci sequence 
having initial conditions F0 = 0, F1 = 1 and recurrence relation Fn = Fn–1 + Fn–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 q0 = 0 and q1 = 1 which is generated by the recurrence relation qn = aqn–1 + qn–2 (when n is even) or qn = bqn–1 + qn–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.
© de Gruyter 2009
Articles in the same Issue
- On the Complexity of n-Player Hackenbush
- On the Equation ax ≡ x (mod b)
- A New Generalization of Fibonacci Sequence & Extended Binet's Formula
- On Rainbow Solutions to an Equation with a Quadratic Term
- 2-Pile Nim with a Restricted Number of Move-Size Imitations
- The Finite Heine Transformation and Conjugate Durfee Squares
- A Short Note on the Difference Between Inverses of Consecutive Integers Modulo p
- Factorization Results with Combinatorial Proofs
- Squares in (12 + m2) ⋯ (n2 + m2)
- On Monochromatic Sets of Integers Whose Diameters Form a Monotone Sequence
- Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins
- Une forme effective d'un théorème de Bateman sur la fonction phi d'Euler
- Differences of Multiple Fibonacci Numbers
Articles in the same Issue
- On the Complexity of n-Player Hackenbush
- On the Equation ax ≡ x (mod b)
- A New Generalization of Fibonacci Sequence & Extended Binet's Formula
- On Rainbow Solutions to an Equation with a Quadratic Term
- 2-Pile Nim with a Restricted Number of Move-Size Imitations
- The Finite Heine Transformation and Conjugate Durfee Squares
- A Short Note on the Difference Between Inverses of Consecutive Integers Modulo p
- Factorization Results with Combinatorial Proofs
- Squares in (12 + m2) ⋯ (n2 + m2)
- On Monochromatic Sets of Integers Whose Diameters Form a Monotone Sequence
- Palindromes in Different Bases: A Conjecture of J. Ernest Wilkins
- Une forme effective d'un théorème de Bateman sur la fonction phi d'Euler
- Differences of Multiple Fibonacci Numbers
