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

Fibonacci numbers