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Questions tagged [fibonacci-numbers]

Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

The $n$th Fibonacci number $F_n$ is defined recursively, by

$$F_n = F_{n - 1} + F_{n - 2}$$

for $n > 1$, and $F_0 = 0,\; F_1 = 1$. There is a closed form expression, namely

$$F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}$$

where the golden ratio $\varphi$ is equal to $\frac{1 + \sqrt{5}}{2}$.

Combinatorial identities involving the Fibonacci numbers have been extensively studied, and the numbers arise frequently in nature and in popular culture.

Reference: Fibonacci number.

2190 questions
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Proof and geometrical significance of $F(n+1)^2-F(n) \cdot F(n+2)$?

My son notes that for Fibonacci numbers $F_n$, $$ (F_{n+1})^2-F_n \cdot F_{n+2} =(-1)^n $$ I assume that this is true. Update: I see that the proof is already here: Prove the given property of the Fibonacci numbers , so never mind about that…
djsadinoff
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How is this identity for Fibonacci numbers called?

In the course of proving another identity, I've found that $$F_n \equiv F_kF_{n-k+1} + F_{k-1}F_{n-k}$$ …for all corresponding n and k. However, this (or something similar) has assuredly already been named. What's the above referred to as?
Columbo
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Fibonacci numbers notation

$F_{i}$ denotes the $i^{th}$ Fibonacci number, but what does it mean when there are 2 subscripts, $F_{ij}$? Context: show that $F_{i}$|$F_{ij}$ (where $i$ and $j$ are positive integers) Thanks
User12
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Fibonacci clarification

As explained in Excursion 4.5, the Fibonacci numbers are defined by the rules: F(0) = 0, F(1) = 1, and for all n with n ≥ 2, F(n) = F(n-1) + F(n-2). Which of these claims about the Fibonacci numbers is false? Select one: a. For every n with n ≥ 5,…
user_123945839432
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Fibonacci numbers relation

I was wondering if there was a relation between a Fibonacci number and its position. Is there a function $f(n)$ such that $$f(n)=F_n$$ where $F_n$ is the nth Fibonacci number?
PunkZebra
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Why does Cassini's identity imply consecutive Fibonacci's number are relatively prime?

From Knuth's The Art of Computer Programming, Volume 1 on page 81 he gives Cassini's identity $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. Relation(4) and follows by saying "Relation(4) shows that $F_n$ and $F_{n+1}$ are relatively prime, since any…
almel
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Proving that a Fibonacci number is divisible by integer a

I am working on a review problem and can't figure out how to go about getting to an answer. We are told to let $F_n$ be the $nth$ Fibonacci number (defined as $F_1=F_2=1,F_{n+1}=F_n+F_{n-1}$). Show that, for any positive integer a, there is some…
user221400
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Prove that the set of solutions to $F_{n+2} = F_{n+1} + F_n$ is of dimension 2

I was playing with the Fibonacci sequence, willing to prove that $$ F(n) = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right) $$ I did the usual, looking for geometric sequences satisfying the…
Jerome
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Why the number of subsets S ⊂ {1,...,n} without an odd number of consecutive integers is F(n+1)?

I have two questions about the Fibonacci sequence: I read from Wikipedia: 1) The number of subsets S ⊂ {1,...,n} without an odd number of consecutive integers is F(n+1). 2) The number of binary strings of length n without an even number of…
user3019105
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Fibonacci inequality

If $F_n$ denotes the $n$-th Fibonacci number ($F_0 = 0, F_1 = 1, F_{n+2} = F_{n+1} + F_n$), show that the inequalities $F_{2n-2} < F_n^2 < F_{2n-1}$ hold for all $n ≥ 3$.
user1321345589
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Adding Fibonacci Numbers

I am getting confused on adding Fibonacci numbers. For example I know that: $\mathrm{F}_\mathrm{K+1}+\mathrm{F}_\mathrm{K}=\mathrm{F}_\mathrm{K+2}$ But I believe my logic is flawed. The way I am thinking about this is that if you add…
woody
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Let $u_n$ be the $n$-th entry in the Fibonacci sequence $1,1,2,3,5,8,13,\ldots$

If you start with $u_1 = 1$ and $u_2 = 1$, then the sequence can be generated using the formula $$u_{n+1} = u_n + u_{n-1}\ .$$ If $u_n = r^n$, what is r? Can anyone figure this out? I am so stuck please help!
user130520
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Help with a proof involving Fibonacci numbers.

I'm working through SICP MIT course, and I'm a little lost on how to prove the following statement. I think I'm able to demonstrate it, but have no idea how to prove this statement. I may misunderstand the statement itself, although I was able to…
stantona
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Generalisation of Fibonacci

Somehow a generalisation of the fibonacci numbers, do numbers created by the formula $ F(n) = F(n-1) + [F(n-1)-F(n-2)+F(n-3)-F(n-4)+F(n-5)-F(n-6).....]$ with $F(1) = 1$ have a specific name?
Paramar
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Fibonacci Sequence and odd/even addition

Prove that f0 – f1 + f2 - … - f2n-1 + f2n = f2n-1 – 1. For n is all positive numbers. I have an idea to what I must do, but I can't figure what the base case is. I think it is f(0) = 0 and f(1) = 1. But then idk how to prove these?
Darneezie
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