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Fungrim entry: 12b336

Fn=enucos ⁣(πn)enu5   where u=log ⁣(φ)F_{n} = \frac{{e}^{n u} - \cos\!\left(\pi n\right) {e}^{-n u}}{\sqrt{5}}\; \text{ where } u = \log\!\left(\varphi\right)
Assumptions:nZn \in \mathbb{Z}
TeX:
F_{n} = \frac{{e}^{n u} - \cos\!\left(\pi n\right) {e}^{-n u}}{\sqrt{5}}\; \text{ where } u = \log\!\left(\varphi\right)

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
FibonacciFnF_{n} Fibonacci number
Expez{e}^{z} Exponential function
ConstPiπ\pi The constant pi (3.14...)
Sqrtz\sqrt{z} Principal square root
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
GoldenRatioφ\varphi The golden ratio (1.618...)
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("12b336"),
    Formula(Equal(Fibonacci(n), Where(Div(Sub(Exp(Mul(n, u)), Mul(Cos(Mul(ConstPi, n)), Exp(Mul(Neg(n), u)))), Sqrt(5)), Equal(u, Log(GoldenRatio))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-19 14:38:23.809000 UTC