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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.

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Fibonaccith fibonacci number

Let $f_n$ denote the $n$th Fibonacci number. A positive integer $n$ is called good if $f_{f_n}$ is divisible by $n$ but $f_n$ is not divisible by $n$. My question is: how many good numbers are there. I think there are infinity many but I can’t prove…
user173628192
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Origin of the constant $\phi$ in Binet's formula of the $n$-th term of the Fibonacci sequence

I have read in a page that " To find the nth term of the Fibonacci series, we can use Binet's Formula" F(n) = round( (Phi ^ n) / √5 ) provided n ≥ 0 where Phi=1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ... . my question…
yuv60
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What is $\lim\limits_{n\to\infty}\sum\limits^n_{i=0}F_{2i}-\phi F_{2i-1}$?

Firstly, the question is: Evaluate $$(1-0\phi)+(2-1\phi)+(5-3\phi)+(8-5\phi)+\cdots=\lim_{n\to\infty}\sum^n_{i=0}F_{2i}-\phi F_{2i-1}$$ where $F_0=1,$ $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for all integer n (even n negative). I attempted the basic…
Wen
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Prove $f_{n+1}^2+f_n^2=f_{2n+1}$

Prove that if $f$ is the Fibonacci sequence then $f_{n+1}^2+f_n^2=f_{2n+1}$ holds for all n. Instead of trying doing this by induction, I need to do it by trying to just replacing the explicit formula for $f_n$. I am stuck at this…
davidaap
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How long is the Zeckendorf Representation of a given number?

Given a number, how does one tell how many Fibonacci numbers will be involved in its Zeckendorf representations (i.e. how many "active" bits)? In base 2, you simply take log-base2 of a number n to get the maximum amount of bits needed to store…
Danyil Bee
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Explaining a Fibonacci

Explain why the number below is not 299th Fibonacci number: 222232244629420445529739893461909967206666939096499764990979600 I need an explanation
Anne
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Interesting Fibonacci sequence pattern

For the equation $x^2+1$ is divisible by $y$ and $y^2+1$ is divisible by $x$ the solutions come out to be alternate terms of the Fibonacci sequence that is (1,2) (2,5) (5,13) (13,34) (34,89) and so on. I would like to know the proof or reason for…
Primepokemon
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Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$

By calculating for $n\in \{1,2,3,4,5,6,7\}$, I've formulated the rule \begin{equation} F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n, \end{equation} where $F_n$ is the $n$th fibonacci number. I want to show that this is true for all $n \in \mathbb{N}$.…
SBS
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Fibonacci identity for $2F_n + F_{n-3}$

For the $n$'th Fibonacci number, I found the following identity: $$ (2F_n + F_{n-3})^2 = 5\cdot F_n^2 + 4\cdot (-1)^n $$ Now, I don't really expect that this is something new, but I'd like to have some kind of explanation, as this identity with its…
Landei
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Does 1/4 eventually repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's 0.131313..., and in base 3 it's just 0.1. A less…
Joe
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How many times a positive number can be found in shifted Fibonacci Sequences?

Given a positive number, how many times can this number be found in shifted fibonacci sequences? ...For example...Number 11, can be present seven times in total, in Fn+3, Fn+6, Fn+8, Fn+9, Fn+10, Fn+10, Fn+11. Can we generalise this? –
Anthony
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Question regarding the Fibonacci sequence

Given the Fibonacci sequence $(F_1, F_2,F_3, ...)$ how do I prove that if $m|n$ then $F_m|F_n$? Can this be proven with mathematical induction?
user176791
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Remarkable relation between Fibonacci numbers and its squares!

There is a remarkable relation between Fibonacci numbers and its squares: $F^{2}_{n} +F^{2}_{n+1}=F_{2n+1}$. I know how to prove it using $F_{n}=\frac{\sqrt{5}}{5}((\frac{1+\sqrt{5}}{2})^n -(\frac{1-\sqrt{5}}{2})^n)$ formula, but I want a proof…
kpax
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Remainder of a high fibonacci number

I found a question in my assessment book: What is the remainder when the 1995th number of the fibonacci sequence is divided by 8? How to solve?
user141870
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New Fibonacci sequence

I have new Fibonacci number That I want to know is there any special direct formula to count $f(n)$. Like the normal Fibonacci: $F(0) = 7$, $F(1) = 11$, $F(n) = F(n-1) + F(n-2)$ (n>=2) For example I want to calculate $f(100)$ without knowing $f(99)$…
Danial
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