Fungrim entry: ad0d7a

F_{n} = \frac{{\varphi}^{n} - \cos\!\left(\pi n\right) {\varphi}^{-n}}{\sqrt{5}}

Assumptions:

n \in \mathbb{Z}

TeX:

F_{n} = \frac{{\varphi}^{n} - \cos\!\left(\pi n\right) {\varphi}^{-n}}{\sqrt{5}}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Fibonacci`	$F_{n}$	Fibonacci number
`Pow`	${a}^{b}$	Power
`GoldenRatio`	$\varphi$	The golden ratio (1.618...)
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("ad0d7a"),
    Formula(Equal(Fibonacci(n), Div(Sub(Pow(GoldenRatio, n), Mul(Cos(Mul(Pi, n)), Pow(GoldenRatio, Neg(n)))), Sqrt(5)))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

Topics using this entry

Fibonacci numbers

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