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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71 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
45 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
2
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1answer
472 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 ...
0
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2answers
233 views

find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
0
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1answer
372 views

Fibonacci proof by Strong Induction

Prove by strong induction that for a ∈ A we have $F_a + 2F_{a+1} = F_{a+4} − F_{a+2}.$ $F_a$ is the $a$'th element in the Fibonacci sequence
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2answers
53 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
85 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 ...
4
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2answers
52 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, ...
0
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0answers
40 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 ...
-1
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2answers
490 views

Fibonacci sequence - how to prove $a_n=\frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$ without induction [closed]

How to prove that $$a_n=\frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$$ without using induction?
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4answers
77 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 $F_i$ ...
1
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1answer
60 views

Fibonacci numbers identity - proof by induction

$\displaystyle F_{k-1} F_{k+1} - F_k^2 = (-1)^k$ I have done the base step for $k=1$ and it works. I realize we need to prove for $k+1$, so: $$F_k F_{k+2} - F_{k+1}^2 = (-1)^{k+1}$$ Could ...
1
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0answers
74 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 ...
0
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2answers
103 views

number theory fibonacci

Using facts of the Fibonacci sequence, I need to show that if $m,n$ are natural numbers that satisfy $m \mid F_n$ and $m \mid F_{n+1}$, then $m=1$. I am not sure where to start with this.. I am ...
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3answers
146 views

Fibonacci divisibility

Is $2051$ a factor of any fibonacci number? It is not a factor of any perfect number. The prime factors of $2051$ are $7$ and $293$, which are both prime. the $8$th fibonacci number, is the first ...
4
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1answer
106 views

Fibonacci numbers expressed as squares of lower Fibonacci numbers

I am no mathematician so my apologies for my ignorance. I notice that every number in the Fibonacci series can be expressed as a previous Fibonacci number squared plus or minus (alternating) another ...
0
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1answer
43 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
79 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 ...
3
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1answer
108 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
65 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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1answer
80 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
117 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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2answers
764 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.
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1answer
418 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
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1answer
59 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
78 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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0answers
173 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
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
48 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 ...
3
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3answers
74 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
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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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4answers
880 views

Conjecture: Only one Fibonacci number is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below ...
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1answer
43 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 $a$) ...
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1answer
293 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 ...
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1answer
102 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 ...
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16answers
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Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
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2answers
82 views

Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
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3answers
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Fibonacci sequence: Given $n$ and $\mathrm{Fib}(n)$, is it possible to calculate $\mathrm{Fib}(n-1)$?

Given $n$ and $\newcommand{\Fib}{\mathrm{Fib}} \Fib(n)$, is it possible to calculate the previous number in the Fibonacci sequence - $\Fib(n-1)$ using integer math in constant time? In other words, I ...
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votes
4answers
149 views

For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$

For the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_n+F_{n+1}$, prove that $$\sum_{i=1}^n F_i= F_{n+2} - 1$$ for $n\ge 1$. I don't quite know how to approach this problem. Can someone help and ...
52
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4answers
3k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
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1answer
36 views

How do I proceed from here on finding the Binet's formula via generating functions?

So, I'm stuck with the algebra for the nth number on the Fibonacci sequence in here. I managed to get to the part where $G(x) = \frac{x}{1-x-x^2}$ $=$ $\frac{x}{(1-\alpha x)(1-\beta x)}$, and I know ...
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2answers
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Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt ...
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3answers
117 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
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2answers
83 views

nth convolved Fibonacci numbers of order 6 modulo m

Problem: Find the coefficient of xk in (1−x−x2)-6 modulo m. Constraints: k≤264 m≤105, m can be a composite number. I have 10^5 such queries to process in 2 sec, so O(log k) for each query ...
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5answers
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Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
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1answer
106 views

Generalised Fibonacci

How to find the nth term of the recurrence in $\log n$ time. $$ \begin{array}{rcl} F[n]&=&F[n-1]+F[n-3]\\ F[2]&=&1\\ F[3]&=&2\\ F[4]&=&3\\ F[5]&=&4 \end{array} ...
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1answer
487 views

Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?

$\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$ (Link), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ \ldots )$ This is ...
15
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4answers
6k views

The generating function for the Fibonacci numbers

Prove that $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$ The coefficients are Fibonacci numbers $\left\{1,1,3,5,8,13,21,...\right\}$.
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0answers
69 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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3answers
75 views

Adding Fibonacci Numbers

I am getting confused on adding Fibonachi 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 ...