# Fungrim entry: b8ed8f

$F_{n} = \frac{1}{{2}^{n - 1}} \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {5}^{k} {n \choose 2 k + 1}$
Assumptions:$n \in \mathbb{Z}_{\ge 0}$
TeX:
F_{n} = \frac{1}{{2}^{n - 1}} \sum_{k=0}^{\left\lfloor \left( n - 1 \right) / 2 \right\rfloor} {5}^{k} {n \choose 2 k + 1}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Fibonacci$F_{n}$ Fibonacci number
Pow${a}^{b}$ Power
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("b8ed8f"),
Formula(Equal(Fibonacci(n), Mul(Div(1, Pow(2, Sub(n, 1))), Sum(Mul(Pow(5, k), Binomial(n, Add(Mul(2, k), 1))), For(k, 0, Floor(Div(Sub(n, 1), 2))))))),
Variables(n),
Assumptions(Element(n, ZZGreaterEqual(0))))

## 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