# Fungrim entry: bceed4

$F_{n} = \frac{\left(1 + \cos\!\left(\pi n\right)\right) \sinh\!\left(n u\right) + \left(1 - \cos\!\left(\pi n\right)\right) \cosh\!\left(n u\right)}{\sqrt{5}}\; \text{ where } u = \log(\varphi)$
Assumptions:$n \in \mathbb{Z}$
TeX:
F_{n} = \frac{\left(1 + \cos\!\left(\pi n\right)\right) \sinh\!\left(n u\right) + \left(1 - \cos\!\left(\pi n\right)\right) \cosh\!\left(n u\right)}{\sqrt{5}}\; \text{ where } u = \log(\varphi)

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Fibonacci$F_{n}$ Fibonacci number
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("bceed4"),
Formula(Equal(Fibonacci(n), Where(Div(Add(Mul(Add(1, Cos(Mul(Pi, n))), Sinh(Mul(n, u))), Mul(Sub(1, Cos(Mul(Pi, n))), Cosh(Mul(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