Fungrim entry: 8a548e

{\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}}^{n} = \begin{pmatrix} F_{n + 1} & F_{n} \\ F_{n} & F_{n - 1} \end{pmatrix}

Assumptions:

n \in \mathbb{Z}

TeX:

{\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}}^{n} = \begin{pmatrix} F_{n + 1} & F_{n} \\ F_{n} & F_{n - 1} \end{pmatrix}

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`Fibonacci`	$F_{n}$	Fibonacci number
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("8a548e"),
    Formula(Equal(Pow(Matrix2x2(1, 1, 1, 0), n), Matrix2x2(Fibonacci(Add(n, 1)), Fibonacci(n), Fibonacci(n), Fibonacci(Sub(n, 1))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

Topics using this entry

Fibonacci numbers

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

2021-03-15 19:12:00.328586 UTC