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Fungrim entry: bceed4

Fn=(1+cos ⁣(πn))sinh ⁣(nu)+(1cos ⁣(πn))cosh ⁣(nu)5   where u=log(φ)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:nZn \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
FibonacciFnF_{n} Fibonacci number
ConstPiπ\pi The constant pi (3.14...)
Sqrtz\sqrt{z} Principal square root
Loglog(z)\log(z) Natural logarithm
GoldenRatioφ\varphi The golden ratio (1.618...)
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("bceed4"),
    Formula(Equal(Fibonacci(n), Where(Div(Add(Mul(Add(1, Cos(Mul(ConstPi, n))), Sinh(Mul(n, u))), Mul(Sub(1, Cos(Mul(ConstPi, n))), Cosh(Mul(n, u)))), Sqrt(5)), Equal(u, Log(GoldenRatio))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

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2019-10-05 13:11:19.856591 UTC