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Fungrim entry: ad0d7a

Fn=φncos ⁣(πn)φn5F_{n} = \frac{{\varphi}^{n} - \cos\!\left(\pi n\right) {\varphi}^{-n}}{\sqrt{5}}
Assumptions:nZn \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
FibonacciFnF_{n} Fibonacci number
Powab{a}^{b} Power
GoldenRatioφ\varphi The golden ratio (1.618...)
ConstPiπ\pi The constant pi (3.14...)
Sqrtz\sqrt{z} Principal square root
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("ad0d7a"),
    Formula(Equal(Fibonacci(n), Div(Sub(Pow(GoldenRatio, n), Mul(Cos(Mul(ConstPi, n)), Pow(GoldenRatio, Neg(n)))), Sqrt(5)))),
    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