Tagged Questions

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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38 views

Mean and Variance of Fibonacci Numbers

I would like to ask the community for feedback regarding the following two conjectures of mine: $\textbf{Conjecture 1}$ Let $\mathcal{F}_N^- = \{F_n:-N \leq n < 0\}$, i.e. be the set of Fibonacci ...
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24 views

What's the point of Fermat (and other) numbers?

The title says it all. I mean anybody can can make up a sequence like Fn = (F(n-1) - 1)^3 + 3 or something like that- why is Fermat (or Fibonacci or whatever) numbers better or more important?
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1answer
27 views
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1answer
24 views

how to calculate a modified fibonacci via matrix exponentiation

If I modify the fibonacci recurrence to be the following way: f(0) = 1 f(1) = 1 f(N) = f(N - 1) + f(N - 2) + 1 Is it possible to represent this recurrence in a matrix equation similar to the one ...
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0answers
16 views

geometric proof for fibonacci numbers identity with sum of two squares

Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares The link above gives the induction proof does a geometric proof using the squares with Fibonacci length exist for this?
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0answers
43 views

Does this sequence contain all positive integers?

Set $a_1 = 1$. Then $a_n$ is chosen to be the smallest distinct positive integer such that $$\frac{\sum_{i = 1}^n a_i}{n}$$ is a Fibonacci number. If my calculations are correct, the sequence starts ...
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2answers
57 views

Use Fibonacci number to prove that is the integer that is closest to another number

Hi everyone, I don't really understand the problem. I have the following hint, but I don't know how to work it.
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0answers
27 views

Proof for $\gcd(F_m,F_n)=F_{\gcd(n,m)}$ [duplicate]

I saw many questions/answers, where: $$\gcd(F_m,F_n)=F_{\gcd(n,m)}$$ is taken as a fact. But how can I actually prove that this is true?
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4answers
82 views

Find the sum of an infinite series of Fibonacci numbers divided by doubling numbers. [duplicate]

How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time ...
2
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2answers
60 views

$\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$

Please help! I need help on my assignment for discrete mathematics! Prove the following identity: $\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$ I need to ...
2
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0answers
19 views

Symmetric Fibonacci images

I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number: Rotate the turtle $k (f_n ...
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0answers
32 views

Proof of a formula for generalized Fibonacci numbers

I have done the verification for $$U_rU_{n−1} − U_{r−1}U_n = (−1)^{r−1}U_{n−r}$$ I realized when I was doing for $n=k+1$, the expression $U_rU_k − U_{r−1}U_{k+1}$ would not equate to ...
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0answers
13 views

Calculating the hitting probability using the strong markov property

We have the following Markov chain. $X_n=(F_{n-1},F_n)$ where $F_0=0, F_1=1$ and with probability 1/2 $F_{n+1}$ is the difference of $F_{n-1}$ and $F_{n}$ and with probability 1/2 the sum. I have to ...
1
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1answer
38 views

How to find $\sum_{n = 0}^{ \infty} \frac{F_n}{3^n}$

How can I find $$\sum_{n = 0}^{ \infty} \frac{F_n}{3^n}$$ If I know that the generating function for the Fibonacci sequence is $G(t) = \frac{t}{1 - t - t^2}$?
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2answers
48 views

Proof a number is Fibonacci number

I have a question regarding the proof that a number n is a Fibonacci number if and only if $5n^2-4$ or $5n^2+4$ is a perfect square. I don't understand the second part of the proof: knowing that ...
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4answers
86 views

Proof of ${F(n+4)}^{4} - {4F(n+3)}^{4} - {19F(n+2)}^{4} - {4F(n+1)}^{4}+{F(n)}^{4} = -6$

Observe: \begin{matrix} F(n)|&{F(n)}^{4}& - {4F(n+1)}^{4}& - {19F(n+2)}^{4}&- {4F(n+3)}^{4}&{F(n+4)}^{4}& = -6\\ 1|& 1& -4& -304& -324& 625&=-6\\ ...
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0answers
30 views

Applying Fibonacci Fast Doubling Identities

So I sort of understand of how these identities came about from reading this article. $F_{2n+1} = F_{n}^2 + F_{n+1}^2$ $F_{2n} = 2F_{n+1}F{n}-F_{n}^2 $ But I don't understand how to apply them. ...
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0answers
37 views

proving Fibonacci numbers using mathematical Induction?

Can anyone confirm whether my answer is correct, please. Let suppose we have the following fibonacci numbers as shown: $f(0) = 0, f(1) = 1$, and $f(n) = f(n-1) + f(n-2)$ for $n \geq 2$. Prove that ...
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1answer
52 views

Prove that for each Fibonacci number $f_{4n}$ is a multiple of $3$. [duplicate]

The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \ge 2$. Prove that for each $n \ge 0$, $f_{4n}$ is a multiple of $3$. I've tried to prove to ...
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2answers
63 views

Determine whether $A215928(n)=G_n$.

Let $G_0=1$ and $G_{n+1}=F_0G_n+F_1G_{n-1}+\cdots+F_nG_0$, where $F_n$ is the $n$th term of the Fibonacci sequence, i.e., $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. Let $P_0=P_1=1,\ P_2=2,$ and ...
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5answers
48 views

Prove by Induction $F_{2n} = F_{n} * L_{n}$, for n >= 1

Where $F$ is the Fibonacci Sequence, and $L$ is the Lucas Sequence. I need to find the inductive proof of this statement. I've got nearly a page of work in front of me trying to use definitions such ...
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2answers
58 views

Mathematical induction used on Fibonacci Sequence

I have no clue how to go about doing this question using induction. It states that the Fibonacci sequence is defined as: F0 = 0 F1 = 1 Fn = Fn-2 + Fn-1 for n>=2 Let S(n) = Fo + F1 + F2 +...+ ...
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3answers
104 views

Fibonacci sequence: how does $0$ get to $1$?

In the Fibonacci sequence, how does $0$ get to $1$? $$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$$ The rule is adding the previous $2$ numbers, and the previous $2$ numbers before $1$ are $0$ and ...
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0answers
55 views

Arctangents, Fibonacci numbers, and the golden ratio

In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice ...
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3answers
37 views

What sequences where the difference between their consecutive terms is always a fibonacci numbers?

What sequence where the difference between its consecutive terms is always a fibonacci numbers ? I am trying to figure out a pattern in this sequence : 1,2,4,7,12,20,33,54,88
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2answers
36 views

Trouble with Fibonacci number mathematical induction

The problem is: $$F_n \leqslant 2F_{n-1}\quad\text{for every integer} \quad n \geqslant 2.$$ I got the smallest case, I just don't know how to get the assumption and the rest of it
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0answers
65 views

Legitimate papers refuting the significance of the golden ratio in art?

I'm not sure this is the right place to ask about this, but is there any legitimate peer-reviewed paper refuting the significance of the golden ratio in art? I can find numerous websites and blogs ...
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2answers
45 views

Number of Fibonacci series that contain a certain integer

In my question, I consider general Fibonacci sequences (sequences satisfying the recurrence relation $F_{n+2}=F_{n+1}+F_n$ independent of their starting value). Given two arbitrary different integers, ...
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0answers
34 views

Sum of reciprocals of (squares of) Fibonacci numbers

What would be the sum of reciprocals of all Fibonacci numbers? What about the sum of reciprocals of squares all Fibonacci numbers? This is not a homework, or something of that sort, the question ...
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2answers
30 views

Relationship between fibonacci number and modulo. [closed]

Determine with reason whether or not $$F_{5n}=0\pmod5$$ I don't have any idea about it.
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4answers
50 views

How to find closed form of summation of Fibonacci Sequence?

I created two formulas to prove a binary theory involving the Fibonacci sequence. (1) $\sum_{i=0}^n F_{2i+1} $ Equation (1) is the sum of all Fibonacci numbers up to $F_n$ where every $i$ in $F_i$ ...
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1answer
70 views

Fibonacci numeration system

Instead of binary or decimal, the Kingdom of Leutonia uses an unusual system to represent numbers, based on the Fibonacci sequence. The Fibonacci sequence $F_0,F_1,F_2,\dots$ is defined recursively ...
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1answer
38 views

Write out the Leutonian numbers that represent the first 12 positive integers.

How could I write out the leutonian numbers that represent the first 12 positive integers ? I have no idea how to start.
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1answer
58 views

determine the number of terms in a fibonacci sequence that are divisible by $3$

Consider the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, . . .$ where each term, after the first two, is the sum of the two previous terms. How many of the first $1000$ terms are divisible by 3?
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2answers
106 views

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?
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1answer
103 views

Powers of 2 in the product of the Fibonacci numbers

I'm working on finding a summation that pulls the powers of 2 out of the product of the Fibonacci numbers. I've noticed some patterns for the Fibonacci number. For example. Looking at the Fibonacci ...
2
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1answer
64 views

Are the Fibonacci numbers' prevalence in nature due to confirmation bias?

The Fibonacci numbers are frequently found in nature, such as in the petals of flowers or the shape of pinecones. But are the numbers actually any more prevalent than other numbers? Could it all be ...
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2answers
716 views

Is there among first 100000001 Fibonacci numbers one that ends with 0000?

This is a difficult problem from competition training: Is there among first 100000001 Fibonacci numbers one that ends with 0000? Trainer says use pigeonhole principle. I do not know how.
2
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1answer
54 views

Proving a Problem involving Fibonacci numbers

I'm working on proving the problem that states $\text {The sequence}$ {$F_n$} $\text {is defined by the} \ F_1=F_2=1, F_{n+2}=F_{n+1}+F_n \ \text {for} \ n \ge 1.$ $\text {For any natural number m, ...
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1answer
73 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
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2answers
76 views

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 ...
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0answers
72 views

Fibonacci sequence: Prove the formula $f_{2n+1}=f_{n+1}^2 + f_n^2$ [duplicate]

I can't seem to figure out this proof. I'm using weak induction and always get stuck during the inductive step. Prove for n > 0: $$f_{2n+1} = f_{n+1}^2 + f_n^2$$
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0answers
40 views

Count fractions between fibonacci

First a little run down on what I am doing: I have a large set of line graph data that I have collected. I've been looking at the graphs first at tick level then for the second chart I combine two ...
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3answers
61 views

When do you need to prove more than 1 base case in the Fibonacci problem?

I was trying to prove that if $F_n = F_{n-1} + F_{n-2}$ and $F_1 = 1$ and $F_2 = 1$, then the following proposition $P(n)$ was true $\forall n : \sum^n_{i=1}F_i=F_{n+2} - 1$ The issue I have with the ...
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0answers
143 views

Prime number distribution theory for dummies

For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis ...
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0answers
27 views

How many rationals for a given $n \in \Bbb N \;\backslash \{1\}$?

Fix $n \in \Bbb N, n> 1$. Now choose a two digit base-$n$ number $ab $ say. There's $n^2$ choices for this. Consider the number $0.c_1 c_2 c_3 \ldots$ where the $c_i$ are defined recursively: ...
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1answer
39 views

Number of strings of size $k$ that do not have 'ab'

Consider $\Sigma = \{a,b,c\}$ and the language $L$, the set of all strings that do not contain 'ab' Find strings, of size $k$ is in $L$ ($L_k$) Consider $A_k$ (strings of size $k$ that end in $a$) ...
2
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1answer
344 views

How many subsets does the set $\{1, 2, \dots , n\}$ have that contain no two consecutive integers if $1$ and $n$ also count as consecutive?

How many subsets does the set $\{1, 2, \dots , n\}$ have that contain no two consecutive integers if 1 and n also count as consecutive? It looks that the number of such subsets obeys the ...
4
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1answer
146 views

Relationship between Primes and Fibonacci Sequence

I recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connection ...
8
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1answer
88 views

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...