Some properties of Fibonacci numbers, Fibonacci octonions, and generalized Fibonacci-Lucas octonions

Some Asymptotic Properties of Generalized Fibonacci Numbers

1. INTRODUCTION Horadam [1] has generalized two theorems of Subba Rao [3] which deal with some asymptotic p r o p e r t i e s of Fibonacci numbers. Horadam defined a sequence {w (n 2) where a , a are the roots of x 2-P 21 x + P 2 2 = 0. We shall let 06 """ LX r\ r\ UO r\-| • Horadam established two theorems for {w n }: I. The number of terms of {w n } not exceeding N is asymptotic to log(Nd/(P ...

Some New Finite Sums Involving Generalized Fibonacci And Lucas Numbers

Some Properties of Fibonacci Numbers

We formalized some basic properties of the Fibonacci numbers using definitions and lemmas from [7] and [23], e.g. Cassini’s and Catalan’s identities. We also showed the connections between Fibonacci numbers and Pythagorean triples as defined in [31]. The main result of this article is a proof of Carmichael’s Theorem on prime divisors of prime-generated Fibonacci numbers. According to it, if we ...

Restricted Permutations, Fibonacci Numbers, and k-generalized Fibonacci Numbers

In 1985 Simion and Schmidt showed that the number of permutations in Sn which avoid 132, 213, and 123 is equal to the Fibonacci number Fn+1. We use generating function and bijective techniques to give other sets of pattern-avoiding permutations which can be enumerated in terms of Fibonacci or k-generalized Fibonacci numbers.

Sums of products of generalized Fibonacci and Lucas numbers

In this paper, we establish several formulae for sums and alternating sums of products of generalized Fibonacci and Lucas numbers. In particular, we recover and extend all results of Z. Čerin [2, 2005] and Z. Čerin and G. M. Gianella [3, 2006], more easily.