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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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
28 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
9 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 ...
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1answer
29 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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0answers
39 views

show $\operatorname{Fib}(n)$ is a primitive recursive function? [closed]

let $0,1,1,2,3,5,8,13,21,..$ and $\operatorname{Fib}(0)=0$ $\operatorname{Fib}(n+1)=\overline{sg}(n)+sg(n)\cdot[\operatorname{Fib}(n\dot{-}1)+\operatorname{Fib}(n)]$ show $\operatorname{Fib}(n)$ is ...
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4answers
85 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
24 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
233 views

Cycles in the Generalized Fibonacci Sequence modulo a Prime

Suppose I have a fibonacci sequence 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 Now if I have a modulo 5 fibonacci sequence,it will look like ...
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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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0answers
61 views

show Fibonacci Sequence is a primitive recursive function? [closed]

question:show Fibonacci Sequence is a primitive recursive function? answer: could proof with Induction? $u_{0}=K_{0}$ and $K_{0}$ is costant function.so it is a a primitive recursive function and how ...
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6answers
255 views

A Non-recursive Fibonacci Sequence

How can I determine the general term of a Fibonacci Sequence? Is it possible for any one to calculate F2013 and large numbers like this? Is there a general formula for the nth term of the Fibonacci ...
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2answers
61 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
46 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
54 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
102 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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1answer
432 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
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0answers
53 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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1answer
326 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 ...
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2answers
226 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 ...
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1answer
267 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
35 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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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
43 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
31 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
439 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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2answers
29 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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1answer
324 views

Construction of generating function from identity

I am trying to solve identity involving binomials and fibbonaci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose ...
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4answers
47 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
56 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 ...
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0answers
58 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 ...
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2answers
101 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
120 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 ...
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1answer
99 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 ...
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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
68 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
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
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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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
104 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
713 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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2answers
75 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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1answer
401 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
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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0answers
139 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
38 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 ...