#### Presentation Title

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#### Presentation Type

Presentation

#### Full Name of Faculty Mentor

Prashant Sansgiry, Mathematics and Statistics

#### Major

Applied Mathematics

#### Presentation Abstract

The Fibonacci sequence has the following recursive relationship: xn+2 = xn+1 + xn with x1 = x2 = 1. The modified relationship we analyze is: xn+2 = imxn+1 + xn √with x1 = x2 = a + bi. Here m is a parameter, a and b are real numbers and i is the complex number −1. These relationships are classified as second order difference equations, and we use techniques from difference equations to analyze the behavior of resulting solutions to this equation. Here we study the relationship for real number values of m. We notice that the recursive relationship is periodic for 2 < m < 2. In this presentation we will explain results related to this behavior. For different values of m we observe that the sequence, when plotted in the complex plane, has a surprising geometric relationship that connects to the conic sections:ellipses and hyperbolas.

#### Location

Room 2 (BRTH 112)

#### Start Date

12-4-2022 3:50 PM

#### End Date

12-4-2022 4:10 PM

#### Disciplines

Applied Mathematics

#### Recommended Citation

Meade, Jensen, "On the periodicity of a modified Fibonacci sequence" (2022). *Undergraduate Research Competition*. 50.

https://digitalcommons.coastal.edu/ugrc/2022/fullconference/50

On the periodicity of a modified Fibonacci sequence

Room 2 (BRTH 112)

The Fibonacci sequence has the following recursive relationship: xn+2 = xn+1 + xn with x1 = x2 = 1. The modified relationship we analyze is: xn+2 = imxn+1 + xn √with x1 = x2 = a + bi. Here m is a parameter, a and b are real numbers and i is the complex number −1. These relationships are classified as second order difference equations, and we use techniques from difference equations to analyze the behavior of resulting solutions to this equation. Here we study the relationship for real number values of m. We notice that the recursive relationship is periodic for 2 < m < 2. In this presentation we will explain results related to this behavior. For different values of m we observe that the sequence, when plotted in the complex plane, has a surprising geometric relationship that connects to the conic sections:ellipses and hyperbolas.