On Euler and Fibonacci Numbers
Why Pi is Bounded by Twice Phi
G.K. van der Wal (TU Delft - Electrical Engineering, Mathematics and Computer Science)
RJ Fokkink – Mentor (TU Delft - Applied Probability)
Y. van Gennip – Graduation committee member (TU Delft - Mathematical Physics)
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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.