# Fungrim entry: 3a9c67

$\begin{pmatrix} F_{n + m} \\ F_{n + m - 1} \end{pmatrix} = {\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}}^{m} \begin{pmatrix} F_{n} \\ F_{n - 1} \end{pmatrix}$
Assumptions:$n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}$
TeX:
\begin{pmatrix} F_{n + m} \\ F_{n + m - 1} \end{pmatrix} = {\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}}^{m} \begin{pmatrix} F_{n} \\ F_{n - 1} \end{pmatrix}

n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; m \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Fibonacci$F_{n}$ Fibonacci number
Pow${a}^{b}$ Power
Matrix2x2$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ Two by two matrix
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("3a9c67"),
Formula(Equal(Matrix2x1(Fibonacci(Add(n, m)), Fibonacci(Sub(Add(n, m), 1))), Mul(Pow(Matrix2x2(1, 1, 1, 0), m), Matrix2x1(Fibonacci(n), Fibonacci(Sub(n, 1)))))),
Variables(n, m),
Assumptions(And(Element(n, ZZ), Element(m, ZZ))))

## Topics using this entry

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