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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Reccurence Relation of Subsets

How many subsets does the set S = {1,2,...,n} have that contain no two consecutive integers? Assuming that the number of subsets is f(n), you need to find a recurrence relation and its initial ...
6
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1answer
147 views

Product of consecutive Fibonacci numbers divisibility

Prove that the product of any $k$ consecutive Fibonacci numbers is divisible by the product of the first $k$ Fibonacci numbers. We can try to show that for every prime $p$, the power of $p$ appearing ...
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2answers
56 views

Fibonacci divisibility

Prove that the following holds: $3|F_n$ if and only if $4|n$ Base case for $n=1$: $F_1$=1, so $F_1$ is not divisible by 3 and 1 is not divisble by 4. So the proposition holds for $k=1$ Continue ...
8
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2answers
665 views

Square Fibonacci numbers

Are there Fibonacci numbers other than $F_0 = 0 = 0^2, F_1 = F_2 = 1 = 1^2,$ and $F_{12} = 144 = 12^2$ which are square numbers? If not, what is the proof?
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3answers
66 views

Fibonacci inequality

If $F_n$ denotes the $n$-th Fibonacci number ($F_0 = 0, F_1 = 1, F_{n+2} = F_{n+1} + F_n$), show that the inequalities $F_{2n-2} < F_n^2 < F_{2n-1}$ hold for all $n ≥ 3$.
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1answer
163 views

Solving a question about Fibonacci and Lucas numbers using induction

Im working on practice problems that the instructor gave us yesterday, and I absolutely have no clue of how to solve this problem.. I need to use mathmatical induction to solve this problem.. The ...
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1answer
60 views

Recurrence Fibonacci Sequence Proof

I'm having troubles proving that in a fibonacci sequence if n is divisible by four, then Fn is divisible by three So when Fn is 6, n is 8 and so on. I was thinking maybe I could use mod 3 or mod 4 ...
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2answers
1k views

Understanding this pattern behind the Fibonacci sequence

To be honest, I'm pretty awful at mathematics however, when up till 6AM I do like to do random things throughout the night to keep me occupied. Tonight, I began playing with the Fibonacci sequence in ...
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3answers
42 views

Uniform Convergence for a sequence of functions defined by recurrence

The following is a problem that I can't solve, and I need some tips: Problem: For $x>-1$, define $f_1(x) = x,\ f_{n+1}=\dfrac{1}{1+f_n(x)}$. Find the limit function $f(x)$ and all the subsets of ...
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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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2answers
49 views

How to show that the limit of a fibonacci sequence equates to 1 as n goes to infinity

$$\lim_{n \to \infty} \frac{f_{n+1} f_{n-1}}{f_n^2} = 1$$ I tried expanding both the numerator and denominator to probably cancel out but that did not work... I also split it up into different ...
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1answer
48 views

Prove $f_{n}^{2} = f_{n-2}f_{n+2}+(-1)^{n}$ $n\ge 3$

Prove $f_{n}^{2} = f_{n-2}f_{n+2}+(-1)^{n}, n\ge 3$ (For the sake of space, I'm going to skip the basis step and move straight to the inductive step.) Inductive Step: Assume P(n) is true, prove ...
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1answer
45 views

Proof of: $f_{n}^{2} = f_{n-1}f_{n+1}+(-1)^{n+1}$

So I'm going over some examples of recursion and Fibonacci Sequences for my quiz tomorrow and I'm a bit lost after a certain point. Prove $f_{n}^{2} = f_{n-1}f_{n+1}+(-1)^{n+1}$ $n\geq 2$ Basis ...
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0answers
36 views

Inductive proof of the property $f(k+2)=f(k)+f(k+1)$ for the numbers given in terms of the golden ratio [duplicate]

Help prove through induction that $f(k+2)=f(k)+f(k+1)$ using the golden ratio $\frac1{\sqrt5}\phi^n-\frac1{\sqrt5}\left(\frac{1-\sqrt5}2\right)^n$ $F_{n+2}=F_n+F_{n+1}$, using golden ration $f_n = ...
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0answers
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 ...
5
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1answer
283 views

How many distinct ways to climb stairs in 1 or 2 steps at a time? (Fibonacci puzzle)

There is a very interesting puzzle for fibonacci sequence ...
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1answer
70 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 ...
2
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2answers
155 views

Show that Fibonacci and Lucas numbers satisfy the following equality for all n ≥ 2.

Fibonacci numbers F1, F2, F3, . . . are defined by the rule: F1 = F2 = 1 and Fk = Fk−2 + Fk−1 for k > 2. Lucas numbers L1, L2, L3, . . . are defined in a similar way by the rule: L1 = 1, L2 = 3 and ...
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0answers
52 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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1answer
68 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
204 views

Prove the $n$th Fibonacci number is the integer closest to $\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n$

Prove that the $n$th Fibonacci number $f_n$ is the integer that is closest to the number $$\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n.$$ Hi everyone, I don't really understand the ...
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0answers
60 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
123 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
128 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 ...
3
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0answers
26 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
38 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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1answer
73 views

Calculating the hitting probability using the strong markov property

** This problem is from Markov Chains by Norris, exercise 1.5.4.** A random sequence of non-negative integers $(F)n)_{n\ge0}$ is obtained by setting $F_0=0$ and $F_1=1$ and, once $F_0,\ldots,F_n$ are ...
2
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1answer
44 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
73 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
100 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\\ ...
2
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2answers
138 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
273 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 ...
0
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1answer
78 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
70 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
49 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
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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
144 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 ...
2
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0answers
91 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
49 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
40 views

Understanding Fibonacci Proof

I'm trying to show that $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2} ∀ ≥1$$ where $$F_k = F_{k-1} + F_{k-2}$$ with $$F_0 = F_1 = 1$$ Let P(n) = $$F^2_{k+1} - F^2_k = F_{k-1} * F_{k+2}$$ Basic Step: ...
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2answers
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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
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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
55 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, ...
2
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4answers
116 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
90 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
52 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
84 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
167 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
115 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
128 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 ...