# Fungrim entry: c4d78a

$F_{n} = \frac{2}{\sqrt{5}} {\left(-i\right)}^{n} \sinh\!\left(n \left(\log(\varphi) + \frac{1}{2} \pi i\right)\right)$
Assumptions:$n \in \mathbb{Z}$
TeX:
F_{n} = \frac{2}{\sqrt{5}} {\left(-i\right)}^{n} \sinh\!\left(n \left(\log(\varphi) + \frac{1}{2} \pi i\right)\right)

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Fibonacci$F_{n}$ Fibonacci number
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
ConstI$i$ Imaginary unit
Log$\log(z)$ Natural logarithm
GoldenRatio$\varphi$ The golden ratio (1.618...)
Pi$\pi$ The constant pi (3.14...)
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("c4d78a"),
Formula(Equal(Fibonacci(n), Mul(Mul(Div(2, Sqrt(5)), Pow(Neg(ConstI), n)), Sinh(Mul(n, Add(Log(GoldenRatio), Mul(Mul(Div(1, 2), Pi), ConstI))))))),
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