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 $F_1=1$.

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3
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3answers
104 views

Has anyone invented a computationally simple method to calculate the probability of at least n1-consecutive die rolls, for n2-sided die, n3-rolls?

I think I have invented a formula that allows a computer to very easily calculate the probability of at least n1-consecutive die rolls, on an n2-sided die, rolling it n3-times. For example, for a 3-...
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0answers
24 views

Fibonacci sequence into explicit using backtracking method [closed]

How can the fibonacci sequence be converted from recursive to an explicit formula using backtracking?
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0answers
25 views

Patterns made of Fibonacci's sequence

I was trying to test music with math, so why not Fibonacci? I'd use it lowering every number of the sequence to 7 or minor, I mean, if a certain number n is greater than 7, I'd do n-1, until getting ...
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0answers
36 views

Is there an integral for the Reciprocal Fibonacci constant?

Inspired by this post, I thought I would ask a similar question. The Reciprocal Fibonacci constant $\psi$ is the value the infinite sum of the reciprocals of the Fibonacci numbers converge to, i.e: $$...
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2answers
101 views

Generalised Fibonacci Sequence & Linear Algebra

Consider a generalised fibonacci $G$ sequence $1, 1, 1, 3, 5, 9, 17...$ that's created by summing the last 3 entries in the sequence together: $G_0 = 1, G_1 = 1, G_2 = 1$ and $G_{n+1} = G_n + G_{n-1} ...
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1answer
34 views

Proving Fibonacci sequence by induction method

I am trying to make a conjecture as to Fibonacci numbers which are divisible by 3 and trying to prove it by mathematical induction where the initial conditons are 0 and 1. My problem is that I ...
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2answers
81 views

Which of the following numbers is a Fibonacci number; $(A) 75023$ $(B) 75024$ $(C) 75025$ $(D) 75026$?

This question appeared in one of the mathematical societies exams in Saudi Arabia. No calculator is allowed. The required time to solve one question is $4$ minutes (on average). There is only one ...
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0answers
50 views

a property of self-Fibonacci numbers

A self-Fibonacci number is a number $n$ such that $n$ divides $Fib(n)$. https://oeis.org/A023172 Does this sequence contain a product of any two its elements? I didn't found this exact result among ...
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0answers
27 views

Fibonacci Numbers Identity ${F_n}^2 + {F_{n+1}}^2 = F_{2n+1}$ [duplicate]

Prove the following identities involving Fibonacci numbers: $${F_n}^2 + {F_{n+1}}^2 = F_{2n+1}.$$ I am not sure how to work with the 2n's.
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2answers
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Relating Fibonacci numbers to binomial coefficient [closed]

Let $F_n$ denote the $n^{\text{th}}$ Fibonacci number, show that $$F_{2n+2}=\sum_{i+j \leq n}\binom{n-i}{j}\binom{n-j}{i}$$
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1answer
45 views

How can I show Fibonacci identity?

How can I Show a Fibonacci identity: $$\sum_{k=0}^{n}F_{k}F_{n-k}=\sum_{k=0}^{n}(k+1)F_{k+1}(-2)^{n-k}$$ Can anyone help me, i have no idea.
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28 views

what is the proper notation to express an dependant recursive formula and can one be solved

Given a sequence of numbers, https://oeis.org/A001906 [0, 1, 3, 8, 21, 55, 144, 377] What would be the proper mathematical way to define this number set, and how would it be resolved explicitly ...
3
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1answer
99 views

Proof by induction - Fibonacci

I have a sequence defined as: $u_1 = 1$ $u_2 = 1$ $u_{k+1} = u_{k-1} + u_k$ Which gives: $$1, 1, 2, 3, 5, 8\ldots$$ I need to prove for $n \geqslant 2$ that: $$u_n^2 = u_{n-1}\cdot u_{n+1} + (-1)...
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2answers
58 views

A conjecture concerning Fibonacci

First of all, I'm aware that this conjecture has probably already been made and proved. I was just playing around, and I'd like to know whether it is in fact true and perhaps a hint as to how to prove ...
3
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1answer
73 views

How to prove the following strange relation concerning fibonacci numbers

Is the following relation concerning fibonacci numbers, $F_n$ true? $$F_{2n-1}^n=2^{2n^2}\prod\limits_{r=1}^{n}\prod\limits_{s=1}^{n}\left(\cos^2\frac{r\pi}{2n+1}+\cos^2\frac{s\pi}{2n+1}\right)$$ I ...
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1answer
25 views

(why) is this ratio the golden ratio?

Looking at a slight variation of the fibonacci sequence f(x) = f(x-1) + f(x-2) + 1 where f(1) = 1, f(2) = 1 I'm trying to find the ratio of this sequence but can't figure out how. To get an ...
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1answer
23 views

Is it natural or the way that the numerical system was invented that these magics with numbers exist in Nature? [closed]

Why do these exhilarating magics with numbers exist in Nature? Is it natural or are they tricks of numbers the way they were invented or explored? Classical Fibonacci Sequence = $0, 1, 0+1=1, 1+1=2, ...
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5answers
166 views

Fibonacci recurrence relation - Principle of Mathematical Induction

The problem: Let $F_n$ be the nth term of the Fibonacci sequence: $F_0 = 0$ $F_1 = 1$ $F_n = F_{n-1} + F_{n-2}$ for $n\geq2$ Prove that $\sum_{i=1}^{n} F_i^2 = F_nF_{n+1}$ for ...
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2answers
78 views

Why do all primes in Fibonacci numbers repeat so regularly

I watched this YouTube video that is mainly about primes as factors of the Fibonacci numbers. It notes that every Fibonacci number after F(12) has a new prime factor not previously seen, and this new ...
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1answer
34 views

Minimum number to be added to n to make it a fibonacci number? [duplicate]

Can we find the minimum number to be added to $ n$ to make it a Fibonacci number? for eg if $n = 6$ then ans $= 2$ since $6 + 2 = 8 $ which is a Fibonacci number
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2answers
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Prove $F_{n-a+1}F_a + F_{n-a}F_{a-1}$

Let $F_n$ denote the nth Fibonacci number, and let $a$ be any integer such that $1 \le a \le n$. Prove that $F_n = F_{n-a+1} F_a + F_{n-a}F_{a-1}$. Pay attention to the base case(s).
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1answer
37 views

Limit of the ratio of consecutive Fibonacci numbers as n approaches negative infinity?

It is known that the lim nβ†’βˆž (𝐹𝑛+1/𝐹𝑛) is the Golden Ratio, however I'm curious to know what this limit would be if n approached negative infinity?
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0answers
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Fibonacci numbers induction proof [duplicate]

I recently found out that the Fibonacci Sequence appears in the Pascal Triangle. If it is written like so: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 etc. If we sum the diagonals, we get: $1 = 1 $ $1 = 1 $ ...
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0answers
56 views

How to solve this sum of a recurrence relation?

I've been given the recurrence relation $a_n = na_{n-1}+n^2a_{n-2}$ where $a_0 = 1$ and $a_1 = 1$. I need to find $a_{1000}$. My understanding is that this is very similar to the Fibonacci sequence. ...
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1answer
44 views

Find solutions of $f(2020)x+f(2019)y=1$ where $f$ is Fibonacci sequence [closed]

I need to find at least one solution of $$f(2020)x+f(2019)y=1$$ with $x,y\in\mathbb{Z}$, where $f(n)$ is the $n^{th}$ Fibonacci number, starting at $f(0)=0$, so that: $$f(0)=0,\qquad f(1)=1,\qquad f(...
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0answers
15 views

Question Involving Golden Search Method and Fibonnaci Search Method theory

Hello, I am really struggling with this question. I (think) I have found the solution to the first constraint for a. Basically that the T(b-a) where 0 less than T less than 0.5 gives me the first ...
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1answer
36 views

Radius of convergence of $\frac{z}{1-z-z^2}$

So I want to find the radius of convergence of $\frac{z}{1-z-z^2}:=\sum_{n=0}^{\infty}F_nz^n$ and also show that $F_n$ are the Fibonacci number. $\underline{\text{My attempt}}$ To show that $F_n$ ...
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1answer
47 views

Prove Fib(n) is closest integer to golden ratio

Does anyone know why it's suggested this proof use $\psi = (1 - \sqrt 5)/2$? I'm a bit lost here. Guidance appreciated. PROOF Link
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1answer
43 views

How to evaluate : $\sum_{n=0}^{\infty}\left(\frac{F_{n+x}}{F_{n+x+1}^2}-\frac{F_{n+x+2}}{F_{n+x+3}^2}\right)$

Thank you to &robjohn for proving this question, I got motivated by it, so I went on to investigate this sum $(1)$, $$S_{x}=\sum_{n=0}^{\infty}\left(\frac{F_{n+x}}{F_{n+x+1}^2}-\frac{F_{n+x+2}}{...
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1answer
43 views

Given a Non-Fibonacci number , find the next Non-Fibonacci number

The Non-Fibonacci sequence is 4,6,7,9,10,11,....... Can we find the next non-fibonacci number if we are given any non-fibonacci number? For example, if n=4 then the answer should be 6 because 6 is ...
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2answers
77 views

Show that: $\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$

How to show that? $$S=\sum_{n=0}^{\infty}(-1)^{n+1}\left(\frac{F_n}{F_{n+1}F_{n+2}}\right)^2=\frac{1}{\phi^3}$$ Where $F_n$ Fibonacci number $F_n=\frac{\phi^n-(-\phi)^{-n}}{\sqrt{5}}$ $$F_n^2=\...
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1answer
24 views

Can we print n distinct natural numbers such that their sum is not a fibonacci number?

Can we print n distinct natural numbers such that their sum is not a fibonacci number? for eg if n = 3 then 3 numbers can be 1, 2, 3 since 1+2+3=6 which is not a fibonacci number
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1answer
72 views

Show that: $\prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi$

How to show that: $$P=\prod_{n=1}^{\infty}\frac{\sqrt{5}F_{2n+2}+1}{\sqrt{5}F_{2n+2}-1}=\phi$$ Where $\phi=\frac{1+\sqrt{5}}{2}$ $$P=\frac{\sqrt{5}F_{4}+1}{\sqrt{5}F_{4}-1}\cdot \frac{\sqrt{5}F_{6}+1}{...
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2answers
75 views

prove that: $\sqrt{2}=\frac{F_n^2+F_{n+1}^2+F_{n+2}^2}{\sqrt{F_n^4+F_{n+1}^4+F_{n+2}^4}}$

Pythagoras's constant in Fibonacci number! How do I show that? $$\sqrt{2}=\frac{F_n^2+F_{n+1}^2+F_{n+2}^2}{\sqrt{F_n^4+F_{n+1}^4+F_{n+2}^4}}\tag1$$ Where $F_n$ is Fibonacci sequence. $$2(F_n^4+F_{...
4
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1answer
64 views

On the ratio $\frac{F_n}{B_n}$

One of the interesting limits that I came up with is: $$\lim_{n\to\infty} \frac{F_{n}}{B_{n}}\;\;\;\;\;\;\;\;\;\; \left( n \in \mathbb N^+\right)$$ Where $F_n$ is the nth Fibonacci number and $B_n$...
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0answers
28 views

About the convergence of $S_n:=\sum_{k=1}^{n}\frac{1}{k^{F_{k}}}$

Consider the following summation: $$S_n:=\sum_{k=1}^{n}\frac{1}{k^{F_{k}}}$$ Where $F_n$ is the nth Fibonacci number. Some of the values are listed below: $$ \begin{array}{r|rr} n&S_n&\...
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2answers
49 views

Sum involving Binomial numbers and Fibonacci numbers

I am trying to prove the following identity using induction: $$ \sum_{j=1}^{n}{n\choose j}F_{2n+1-j}=F_{2n+1}-1 $$ I tried expanding the sum using Pascal's identity (or Stifel's relation) and the ...
2
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2answers
98 views

Where is the flaw in this proof by induction?

I had a task to prove undermentioned property of the Fibonacci series. $$ a_2+a_4+a_6+...+a_{2n}=a_{2n+1}-1$$ where $a_1=1, a_2=1, a_3=2$. For $n=3$: $$a_2+a_4+a_6=1+3+8=12=a_7-1$$ Now let's ...
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1answer
39 views
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1answer
76 views

Calculate a large element of the Fibonacci sequence

I am participating in a challenge where I have to calculate the first four and last four digits of the Fibonacci sequence. My first try was just using $x_n = x_{n-1} + x_{n-2}$ but that takes to long ...
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1answer
79 views

Is there a closed form for the nth Fibonacci number which only involves integer operations?

The classical closed form of the nth fibonacci number is $$x_{n}=\frac{(1+5^{1/2})^{n} -(1-5^{1/2})^n}{\sqrt{5}2^n}$$ But is there any way to compute it with only ...
2
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2answers
183 views

Is my proof of the Fibonacci sequence correct?

Been working on this for some time now but have no idea if it's correct! Any hints are appreciated. Recall the Fibonacci sequence: $f_1 = 1$, $f_2 = 1$, and for $n \geq 1$, $f_{n+2} = f_{n+1} + f_n$. ...
4
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1answer
68 views

Randomize Fibonacci sequence

We all know that if $a_n$ if the $n^{th}$ Fibonacci number we can say:$$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}=Ο†$$ Now let's build another sequence like that:$b_1=b_2=1$ and $b_{n+1}=b_n+b_{n-1}$...
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1answer
33 views

Finding closed form of Fibonacci Sequence using limited information

I'm trying to find the closed form of the Fibonacci recurrence but, out of curiosity, in a particular way with limited starting information. I am aware that the Fibonacci recurrence can be solved ...
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0answers
38 views

Convergence of Dirichlet series related to Fibonacci numbers

Let $z(n)$ be the rank of apparition in the Fibonacci sequence, i.e., $z(n)$ is the smallest positive integer $k$ such that $n$ divides $F_k$. A well-known result is that $z(n)\leq 2n$, for all $n\geq ...
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1answer
22 views

Summation Proof involving the Fibonacci Sequence

I'm working through the chapter 1 exercises of "Data Structures and Algorithm Analysis in Java" 3rd edition by Mark Allen Weiss. Exercise 11 a) in chapter 1 asks to prove the following: $\sum_{i=1}^{...
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2answers
66 views

given Fibonacci series $a_n$ show that $\frac{1}{1-x-x^2}=\sum^{\infty}_{n=1}a_nx^n$ [duplicate]

given Fibonacci series $a_n$ show that $\frac{1}{1-x-x^2}=\sum^{\infty}_{n=1}a_nx^n$ I believe it's related to the fact that $x^2+x=1$
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1answer
44 views

Alternate definitions of 'Fibonacci-like' sequences

The Fibonacci sequence begins $F_1,F_2 = 1$ with the recurrence relation $F_n = F_{n-1} + F_{n-2}$. Alternatively, we may say $F_0 = 0$ and $F_1 = 1$ with the same recurrence relation, and obtain the ...
0
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2answers
54 views

Fibonacci's rabbits variation

In this variation on Fibonacci rabbits the growth of mature rabbits in a period has to be less than 10%. After one period the rabbits are called "young rabbits", after two periods the rabbits will ...
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0answers
48 views

Combinatorial interpretation for sum of powers (specially power=$3$) of Fibonacci numbers

Consider the following sum: $$\sum_{k=0}^{n}F_k^2=F_nF_{n+1}$$ Where $F_n$ is the nth Fibonacci number. Using a little of algebra it's easy to show that the sum telescopes to $F_nF_{n+1}$. Now ...

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