Fungrim entry: 12b336

$F_{n} = \frac{{e}^{n u} - \cos\!\left(\pi n\right) {e}^{-n u}}{\sqrt{5}}\; \text{ where } u = \log(\varphi)$
Assumptions:$n \in \mathbb{Z}$
TeX:
F_{n} = \frac{{e}^{n u} - \cos\!\left(\pi n\right) {e}^{-n u}}{\sqrt{5}}\; \text{ where } u = \log(\varphi)

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Fibonacci$F_{n}$ Fibonacci number
Exp${e}^{z}$ Exponential function
Cos$\cos(z)$ Cosine
Pi$\pi$ The constant pi (3.14...)
Sqrt$\sqrt{z}$ Principal square root
Log$\log(z)$ Natural logarithm
GoldenRatio$\varphi$ The golden ratio (1.618...)
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("12b336"),
Formula(Equal(Fibonacci(n), Where(Div(Sub(Exp(Mul(n, u)), Mul(Cos(Mul(Pi, n)), Exp(Mul(Neg(n), u)))), Sqrt(5)), Equal(u, Log(GoldenRatio))))),
Variables(n),
Assumptions(Element(n, ZZ)))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC