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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7
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182 views

when $F_n^2+F_m^2$ is a square for fibonacci numbers

This is a curiosity question I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem: For which couple of integers $(n,m)$ the ...
7
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504 views

A conjecture about Lucas series

Let $L_n$ be the Lucas series: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}(n>1).$ $p$ is a prime number and $p\equiv3,7\pmod {20}$, hence $\exists x,y\in \mathbb Z:2p=x^2+5y^2.$ Is it true that ...
6
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0answers
114 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 ...
5
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75 views

Show that for a given $s$ there are a finite number of Fibonacci number of form $n^2+s$

It is well known that the last Fibonacci number $F_k$ such that $\exists \ n \in \Bbb{N} : F_k = n^2$ is $144$. Thus there are only $4$ perfect squares among the Fibonacci sequence (assuming you ...
4
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71 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
4
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38 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
4
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56 views

Finding first n so nth fibonacci is c modulo p

This is a question I stumbled upon in an online programming contest archive. The problem statement is simple, given $c \equiv F(n)$ mod $P$ and $P$, where $P$ is a prime of form 5$k$ + 1 or 5$k$ - 1, ...
4
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0answers
97 views

Are there infinite Fibonacci primes if and only if there are infinite Fibonacci numbers that are Fibonacci pseudoprimes?

One of the open questions about the Fibonacci numbers is if there are infinite prime numbers inside the Fibonacci sequence. I wonder if a good approach would be trying to know first if there are ...
4
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101 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on Fibonacci quadratic residue: $F(x)^2 mod F(y)$ = { if y is even: $(F(|x-y|)^2$ } { if y is odd: $-(F(|x-y|)^2$ } for ...
4
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107 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows ...
3
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0answers
64 views

Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ...
3
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0answers
150 views

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$?

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$? By better comparison series than $\sum_0^\infty2^{-k}$ we mean a series $\sum c_k$ s.t. ...
3
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0answers
66 views

Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $ a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, ...
3
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0answers
25 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 ...
3
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388 views

Fibonacci and Lucas numbers related identities

We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following ...
3
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172 views

Which starting conditions for the Fibonacci sequence, gives most primes

I found the following question (at http://aperiodical.com/2012/05/matt-parkers-twitter-puzzle-25-may/): If you start the Fibonacci sequence 2,1 instead of 1,1 do you get more or fewer primes? ...
2
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40 views

Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
2
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89 views

Proving $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $ for Fibonacci numbers

How can I prove the following identity about the Fibonacci numbers by using matrices or determinants? $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $
2
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181 views

LCM of Fibonacci numbers

$\newcommand{\lcm}{\operatorname{lcm}}$There is a nice property of Fibonacci numbers which says that: $$\gcd(F_{a_1}, \ldots, F_{a_n}) = F_{\gcd(a_1, \ldots, a_n)}$$ I am curious is there anything ...
2
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57 views

Generalizing the Fibonacci identity $F_{2n}=-F_{n-1}^2+F_{n+1}^2$

Using an integer relations algorithm, we get, $$F_{2n}=-F_{n-1}^2+F_{n+1}^2$$ $$6F_{4n}= -F_{n-2}^4-3F_{n-1}^4+3F_{n+1}^4+F_{n+2}^4$$ The pattern of the subscripts is clear. Expressing the ...
2
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0answers
71 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
2
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0answers
90 views

Who First Considered This Generalization of the Fibonacci Numbers?

I am looking for the author who originally researched a generalization of the Fibonacci numbers, which Koshy, in Chapter 7 his book Fibonacci and Lucas Numbers with Application refers to as the ...
2
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90 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 ...
2
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0answers
90 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
2
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117 views

Fibonacci applied to human population living to dead ratio problem

If this forum is not the right one for my question, please redirect it. I do not know where to ask it. The question might seem tongue-in-cheek, believe me it's not! Last week to occupy my mind, I ...
2
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0answers
71 views

Could Fibonacci numbers be related to Riemann zeros?

this is my question can tghe fibonacci numbers $$ F_{n+2} =F_{n+1} +F_{n} $$ be related to the zeros of the Riemann zeta function ?? i heard that in the webpage ...
2
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0answers
63 views

Summation involving Fibonacci numbers

Find: $$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n} $$ where $F_n$ is $n$-th Fibonacci number.
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93 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
2
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0answers
100 views

Lucas Number Equivalent of the Pisano Period?

I'm doing some work with the Pisano Period, and it's leading to also talk about the Lucas numbers $mod \ m$, and their period. Is there a specific notation already designated? Sticking with using a ...
2
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0answers
110 views

Why is the reciprocal of the second Fibonacci number negative?

The second Fibonacci number is 1, so it's reciprocal should be 1, right? Why is it that I get $-1$ when I plug in $2$ for n in the reciprocal of Binet's equation ...
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51 views

Proving that the g.c.d of non-consectuive Fibonnaci numbers is also a Fibonacci number

I'm trying to prove that: for non-consecutive Fibonacci numbers, and I know that consecutive Fibonacci numbers are co prime, but I just don't how to prove this using what I know. **EDIT: Lulu has ...
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0answers
39 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
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79 views

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 ...
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0answers
62 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
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0answers
63 views

Sum of Power of Two Fibonacci reciprocals

Evaluate $$\sum_{k=0}^{\infty} \frac{1}{F_{2^n}} \;.$$ I'm thinking of using a relation from a term to another.
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26 views

Hankel determinant involving Fibonacci numbers

Let $F_n$ denote the nth Fibonacci number, with $F_1 = F_2 = 1$. Denote by M(n) the nxn Hankel matrix with $i,j $ entry $F_{i+j-1}^{n-1}$, where i and j range from 1 through n. Finally, let d(n) = ...
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0answers
49 views

Proving an equation dealing with Fibonacci numbers

Prove that: $f(2 \cdot k) = f(k) \cdot  f(k + 1) + f(k - 1)  \cdot f(k) $ Where $f(k)$ is the kth Fibonacci number. Also prove that: $f(2 \cdot  k + 1) = f(k) \cdot f(k) + f(k + 1) \cdot f(k + 1) ...
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76 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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0answers
243 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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0answers
32 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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0answers
71 views

Sum of a Sequnce

How to find this sum $$ \sum_{i+j+k=n} \ G_i \cdot G_j \cdot G_k \ for \ all \ i,j,k > 0, $$ $$ G_i = i \cdot F_i, $$ where Fi - ith number Fibonacci, F0=0, F1=1
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116 views

Linear combination of numbers: Express the $n^{th}$ Fibonacci number in terms of known constants.

Given the concept that any number can be expressed as a combination of other numbers can the $n^{th}$ Fibonacci number be expressed in terms of $\zeta(3)$ and $\ln(2/e)$ ? If possible, show all work ...
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499 views

Calculating Pisano periods for any integer

I recently stumbled across this SPOJ question: http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the Pisano period of a number. After I researched my way through the web, I found ...
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0answers
82 views

Simplify Fibonacci Power Series

I am working on an algorithm to count the number of models for Exactly One in Three SAT (X3SAT) instances. It is known that a chain of X3SAT clauses of length $c$ has $F(c+3)$ satisfying assignments ...
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169 views

Is this “Elegant” algorithm for logarithm by Zeckendorf representation, the same as an 'efficient' algorithm?

The algorithm here which computes the exponent $b$ given a base $a$, and given $n$ = $a$^$b$, appears no better to me than simply counting the number of times we divide $n$ through by the base $a$ ...
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0answers
155 views

The zeta-function of Fibonacci sequence?

I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some ...
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0answers
51 views

How to prove a claim about Fibonacci sequence

I have to prove that for any natural number $n$ there exists $i>0$ such that $n\mid F_i$, where $F_i$ is the $i$-th Fibonacci number.
0
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0answers
64 views

A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
0
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0answers
41 views

Problem about fibonacci sequence via quadratic roots in gelfand's algebra text.Need hints.

I have solved a preceding question proving that the common ratio of such a sequence is $ \frac {1+\sqrt{5}}{2} $ or $ \frac {1-\sqrt{5}}{2} $ (resolving a quadratic equation) . The present problem is ...
0
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0answers
26 views

Can't find ANY golden ratio in the schroder house…

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to ...