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Fungrim entry: 3a9c67

(Fn+mFn+m1)=(1110)m(FnFn1)\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:nZandmZn \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
FibonacciFnF_{n} Fibonacci number
Powab{a}^{b} Power
Matrix2x2(abcd)\begin{pmatrix} a & b \\ c & d \end{pmatrix} Two by two matrix
ZZZ\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))))

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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