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

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

On k-Fibonacci numbers of arithmetic indexes

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

A. G. Shannon

Papers

Generalized Fibonacci Number Triples