TITLE:
On the k–Lucas Numbers of Arithmetic Indexes
AUTHORS:
Sergio Falcon
KEYWORDS:
k–Fibonacci Numbers; k–Lucas Numbers; Generating Function
JOURNAL NAME:
Applied Mathematics,
Vol.3 No.10,
October
11,
2012
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.