We introduce a general Fibonacci sequence that generalizes, between others, both the classic Fibonacci sequence and the Pell sequence. These general kth Fibonacci numbers { F k , n } n = 0 ∞ were found by studying the recursive application of two geometrical transformations used in the well-known four-triangle longest-edge (4TLE) partition. Many properties of these numbers are deduce directly from elementary matrix algebra.

On the Fibonacci k-numbers

The general k-Fibonacci sequence { F k , n } n = 0 ∞ were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.

The k-Fibonacci sequence and the Pascal 2-triangle

Among numerical sequences, the Fibonacci numbers Fn have achieved a kind of celebrity status. Indeed, Koshy gushingly refers to them as one of the "two shining stars in the vast array of integer sequences" [16, p. xi]. The second of Koshy's "shining stars" is the Lucas numbers, a close relative of the Fibonacci numbers, about which we will say more below. The Fibonacci numbers are famous for possessing wonderful and amazing properties. Some are well known. For example, the sums and differences of Fibonacci numbers are Fibonacci numbers, and the ratios of Fibonacci numbers converge to the golden mean. Others are less familiar. Did you know that any four consecutive Fibonacci numbers can be combined to form a Pythagorean triple? Or how about this: The greatest common divisor of two Fibonacci numbers is another Fibonacci number. More precisely, the gcd of F, and Fm is Fk, where k is the gcd of n and m.

The fibonacci numbers-exposed

The k-Fibonacci polynomials are the natural extension of the k-Fibonacci numbers and many of their properties admit a straightforward proof. Here in particular, we present the derivatives of these polynomials in the form of convolution of k-Fibonacci polynomials. This fact allows us to present in an easy form a family of integer sequences in a new and direct way. Many relations for the derivatives of Fibonacci polynomials are proven. � 2007 Elsevier Ltd. All rights reserved.

On k-Fibonacci sequences and polynomials and their derivatives

Abstract 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.

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

A Generalized Fibonacci Sequence

where I=2(p-qb), m=2(p-gqa), a= l (1 + \15), b-(1-\/5). The purpose of this article is to find a connection between generalized Fibonacci numbers and Pythagorean number triples. By a Pythagorean (number) triple is meant a set of three mutually prime integers u, v, w for which u2+v2 = w2. The problem to be solved is this: Given such a triple u, v, w, can we find n, p, q such that the integers whose squares appear in (3) below are these u, v, w? The answer is yes. Viewed in this light, Pythagorean triples may be called Fibonacci (number) triples.

Fibonacci Number Triples

(1973). Generalized Fibonacci Number Triples. The American Mathematical Monthly: Vol. 80, No. 2, pp. 187-190.

A. F. Horadam

Papers

A Generalized Fibonacci Sequence

Fibonacci Number Triples

Generalized Fibonacci Number Triples