## Honors Theses

### On a Generalization of Lucas Numbers

2019

Mathematics

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.

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