Honors Theses
Date of Award
2019
Document Type
Undergraduate Thesis
Department
Mathematics
First Advisor
Sandra Spiroff
Relational Format
Dissertation/Thesis
Abstract
In this paper, we consider a generalization of Lucas numbers. Recall that Lucas numbers are the sequence of integers defined by the recurrence relation: L_n = L_{n−1} + L_{n−2} with the initial conditions L_1 = 1 and L_2 = 3(or L_0 = 1 and L_1 = 3 if the first subscript is zero). That is, the classical Lucas number sequence is 1, 3, 4, 7, 11, 18, .... The goal of the present paper is to study properties of certain generalizations of the Lucas sequence. In particular, we consider the following generalizations of the sequence: l_n = al_{n−1} + l_{n−2} if n is even; bl_{n−1} + l_{n−2} if n is odd, for n = 3,4,5,..., where a and b are any nonzero real numbers, with the initial conditions l_0 = 1 and l_1 = 3 (see Section 2.0.1) and l_n = (−1)^nl_{n−1} + l_{n−2} for n = 3, 4, 5, ... with the initial conditions l_1 = 1 and l_2 = 3 (see Section 3.1.2). More precisely, we will determine the generating function and a Binet-like formula for {ln}^∞_{n=0} and demonstrate numerical simulations for {ln^∞_{n=1}, proving some relations using Principle of Mathematical Induction.
Recommended Citation
Irby, Skylyn Olyvia, "On a Generalization of Lucas Numbers" (2019). Honors Theses. 1136.
https://egrove.olemiss.edu/hon_thesis/1136
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