On the k–Lucas Numbers of Arithmetic Indexes

Sergio Falcon

doi:10.4236/am.2012.310175

Abstract

In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.

Highlights

We study the k-Lucas numbers of arithmetic indexes of the form an + r, where n is a natural number and r is less than r
We prove a formula for the sum of these numbers and the sums of the first k-Lucas numbers, and for the even and the odd k-Lucas numbers
We prove these same formulas for the alternated k-Lucas numbers

Summary

Introduction

Let us remember the k-Lucas numbers Lk,n are defined [1] by the recurrence relation Lk,n 1 k Lk,n Lk,n 1 with the initial conditions Lk,0 , Lk,1 k. 2 k k2 4 2 the characteristic roots of the recurrence equation r2 k r 1 0. If we apply iteratively the equation Lk,n 1 k Lk,n Lk,n 1 we will find a formula that relates the k–Lucas numbers to the k–Fibonacci numbers: Lk ,n Lk ,n p 1 Fk , p Lk ,n p Fk , p 1. This formula is similar to the Convolution formula for the k–Fibonacci numbers Fk,n m Fk,n 1Fk,m Fk,n Fk,m 1 [2,3].

On the k-Lucas Numbers of Arithmetic Index

On the k-Fibonacci Numbers of Indexes n and the k-Lucas Numbers