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Fungrim entry: d454a3

∑k=0n(nk)2kFk=F3n\sum_{k=0}^{n} {n \choose k} {2}^{k} F_{k} = F_{3 n}
Assumptions:n∈Z≥0n \in \mathbb{Z}_{\ge 0}
TeX:
\sum_{k=0}^{n} {n \choose k} {2}^{k} F_{k} = F_{3 n}

n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sum∑nf(n)\sum_{n} f(n) Sum
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
FibonacciFnF_{n} Fibonacci number
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("d454a3"),
    Formula(Equal(Sum(Mul(Mul(Binomial(n, k), Pow(2, k)), Fibonacci(k)), For(k, 0, n)), Fibonacci(Mul(3, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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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