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Fungrim entry: fd732d

Fn=25{sinh⁡ ⁣(nu),n evencosh⁡ ⁣(nu),n odd   where u=log⁡(φ)F_{n} = \frac{2}{\sqrt{5}} \begin{cases} \sinh\!\left(n u\right), & n \text{ even}\\\cosh\!\left(n u\right), & n \text{ odd}\\ \end{cases}\; \text{ where } u = \log(\varphi)
Assumptions:n∈Zn \in \mathbb{Z}
TeX:
F_{n} = \frac{2}{\sqrt{5}} \begin{cases} \sinh\!\left(n u\right), & n \text{ even}\\\cosh\!\left(n u\right), & n \text{ odd}\\ \end{cases}\; \text{ where } u = \log(\varphi)

n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
FibonacciFnF_{n} Fibonacci number
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("fd732d"),
    Formula(Equal(Fibonacci(n), Mul(Div(2, Sqrt(5)), Where(Cases(Tuple(Sinh(Mul(n, u)), Even(n)), Tuple(Cosh(Mul(n, u)), Odd(n))), Equal(u, Log(GoldenRatio)))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

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2021-03-15 19:12:00.328586 UTC