On Euler and Fibonacci Numbers

Title

On Euler and Fibonacci Numbers: Why Pi is Bounded by Twice Phi

 Author

van der Wal, Gwyn (TU Delft Electrical Engineering, Mathematics and Computer Science)

Contributor

Fokkink, R.J. (mentor)
van Gennip, Y. (graduation committee)

Degree granting institution

Delft University of Technology

 Date

2020-09-11

 Abstract

In this report, we will look at the connection between the Fibonacci and Euler numbers. By using a combinatorial argument including the Fibonacci and Euler numbers, we will prove our main theorem:  Fn·En ≥ n! From the main theorem and the asymptotics of these numbers, we will conclude that π ≤ 2ϕ. We follow the proof in the article of Alejandro H. Morales, Igor Pak & Greta Panova, but we will give a more detailed proof and some extra facts about the Golden Ratio, ϕ, and the Fibonacci and Euler numbers. Finally, the article discusses the number of linear extensions of certain partially ordered sets, or posets. We see that there exist a two-dimensional poset Un and complement poset Ûn, both with n elements, such that the number of linear extensions are respectively En and Fn. We conclude that the Fibonacci and Euler numbers are related to each other.    


 Subject

Golden ratio
Fibonacci numbers
alternating permutations
Euler numbers
asymptotics
partially ordered sets
linear extensions

 To reference this document use:

http://resolver.tudelft.nl/uuid:5094371f-46c8-4963-8396-010ecaf11b93

 Part of collection

Student theses

 Document type

bachelor thesis

 Rights

© 2020 Gwyn van der Wal
Files
PDF BEPVerslag.pdf 4.39 MB
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