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

Fibonacci numbers