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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1answer
27 views

Fibonacci n-step numbers

I have searched the web for a definition and I found that this are typically defined as $F^n_{m} = F^n_{m-1} + F^n_{m-2} + \dots + F^n_{m-n}$ where $n$ stands for the number of previous numbers who ...
0
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1answer
35 views

Reference request for a divisibility property of Fibonacci numbers

Define the Fibonacci numbers $F_n$ by $F_n=F_{n-1}+F_{n-2}$ and initial values $F_0=0$ and $F_1=1.$ I would like to get a reference for the following result: If $p$ is a prime number with $p \equiv ...
7
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2answers
106 views

Why do ratios of these Fibonacci-type sequences approach $\pi$?

Define $A_n$ by $A_1=12$, $A_2=18$, and $A_n=A_{n-1}+A_{n-2}$ for $n\ge3$. Similarly define $B_n$ by $B_1=5$, $B_2=5$, and $B_n=B_{n-1}+B_{n-2}$ for $n\ge3$. Terms of $A_n$: $12, 18, 30, 48, ...
2
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2answers
62 views

Prove that for all $n \geq 1$, $F_{-n}$ = $(-1)^{n+1}F_n$ where F is the Fibonacci numbers.

Prove that for all $n \geq 1$, $F_{-n}$ = $(-1)^{n+1}F_n$ where F is the Fibonacci numbers. I've already shown that the formula holds for $n = 1$ and $n = 2$. So I supposed the formula holds for $n$ ...
0
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1answer
33 views

Sum of digits of Fibonacci number a perfect square

During my problem solving with Fibonacci numbers following thought crossed my mind. How many Fibonacci numbers are there such that sum of its digits is a perfect square? Here is a list of ...
0
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1answer
53 views

Find a generating function with Fibonacci

$$G(x) = \sum_{n=1}^\infty na_n x^n $$ Hello. I need to find a generating function for the summation above, where $a_n$ is the Fibonacci sequence. I have found the generating function for the ...
1
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4answers
64 views

Basic Discrete Mathematics Recurrence question

Good afternoon, I've been assigned the following problem from my Intro to Discrete Mathematics: Show that $\sum_{i=1}^n$ F(i) = F(n+2) - 1 note: F(n) is the nth term in the fibonacci sequence. ...
0
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1answer
104 views

Greatest Common Divisor with Fibonacci Numbers [duplicate]

Prove that for all integers $n\geq 0$: $$\gcd(F_{n+1},F_n)=1$$ I am extremely lost. Please can some provide some hint or direction? Thank you so very much
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4answers
44 views

Fibonacci Numbers Induction?

Show that $a_n=n^2+n+1$ satisfies \begin{cases} a_0=1\\ a_k=a_{k-1}+2k & \text{for $k>0$} \end{cases} I want to use induction to solve this problem. but I don't know what my base will be ...
0
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1answer
46 views

Fibonacci-like formula for Padovan sequence

For the Fibonacci sequence, one can show the following and easy to calculate equation : $$\forall n\in \mathbb Z,~\mathcal F_n=\mathcal F_{\lfloor\frac{n}{2}\rfloor+1}^2-(-1)^n\mathcal ...
0
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1answer
49 views

Prove fibonacci with matrixes [duplicate]

I have a question which i could not figure out the answer to, it was the hardest of them all that i got and i couldnt figure it out, its a proof of fibonaccis serie using matrixes and i need som help ...
2
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3answers
57 views

Proof with Fibonacci Sequence

I was working on my homework assignment for one of my classes and I have come across a proof question that my classmates and I are finding difficult to answer. The problem is asking us to prove that ...
1
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1answer
31 views

How to show this Fibonacci identity? $f_{3n}=f^3_{n+1} + f^3_n - f^3_{n-1}$

I already know that $f_{n+m}=f_{n-1}f_m + f_nf_{m+1}$. By letting $m=n$ it immediately follows that $f_{2n}=f_{n}(f_{n+1} + f_{n-1})$ and from that we get $f_{2n}=f^2_{n+1} - f^2_{n-1}$. From this ...
4
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1answer
256 views

Fibonacci sequence in the factorization of certain polynomials having a root at the Golden Ratio

I was playing around with the Golden Ratio $\Phi = \frac{1 + \sqrt 5}{2}$ on Wolfram Alpha and I noticed that if $F_n$ denotes the $n{th}$ Fibonacci number, then the polynomial $P_n(x) = x^n - F_n x - ...
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0answers
27 views

How to make inductive step for a Fibonacci proof [duplicate]

I have to prove $F^2_{n−1} = F^2_n + F^2_{n−1}$ for any $n >=1$ by induction (for the Fibonacci sequence). For the basis step, I have: $n = 1; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$ $ ...
0
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2answers
60 views

Let $F_n$ denote the nth Fibonacci number and prove that the following re true for every possible integer $n$

$$\sum_{i = 1}^n F_{i}^2 = F_n F_{n+1}$$ -I solved a similar Fibonacci sequence that was the following: $$\sum_{i = 1}^n F_i = F_{n + 2} - 1$$ But, I am having trouble with this one, any help is ...
2
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0answers
25 views

Prove that $\sum^{n}_{i=0}\binom{n}{i}F_{i}=F_{2n}$ [duplicate]

I am asked: Let $F_{i}$ denote the $i$-th Fibonacci number. Prove that $$\sum^{n}_{i=0}\binom{n}{i}F_{i}=F_{2n}$$ I have the base case and the inductive hypothesis, but I'm not sure what ...
12
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1answer
345 views

How to prove this series about Fibonacci number: $\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$? [duplicate]

How to prove this series: I have no idea where to start. $$\sum_{n=1}^{\infty }\frac{F_{n}}{2^{n}}=2$$ where $F_{1}=1,~F_{2}=1,~F_n=F_{n-1}+F_{n-2},~~n\geq 3$.
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0answers
58 views

Fibonacci numbers properties

I've verified that $F_{41} \mod F_{32}= F_{23}$ where $41-32=9$ and $32-9=23$. I suppose these facts are correlated. Is there a simple way to show how? Simpler question: how can I justify that ...
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0answers
52 views

Help me find the mistake in my solution of the limit

Let $x_1=1, x_2=2,$ $$x_n=x_{n-1}+x_{n-2}, (n>2)$$ The task is to find: $$\lim \limits_{x\to\infty}\frac{x_{n+1}}{x_n}!$$ My attempt at solution: We write the recursive formula as: ...
4
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1answer
25 views

First Fibonacci Number with Given Remainder

I wonder is there more effective algorithm than brute-force-search to find the first Fibonacci number with given remainder $~~r~~$ modulo given integer $~~m~~$. $$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
3
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3answers
89 views

Induction Proof for $F_{2n} = F^2_{n+1} - F^2_{n-1}$

As stated in the tag, I'm trying to prove by induction the claim $F_{2n} = F^2_{n+1} - F^2_{n-1}$, where $F_{n}$ is the $n^{th}$ Fibonacci number. I've spent hours on the inductive step without ...
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2answers
191 views

Rate of Convergence vs Radius of Convergence

What is the difference between finding the 'rate of convergence' and the radius of convergence'? The question I am trying to solve here is to find the rate of convergence of the ratio of Fibonacci ...
2
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1answer
67 views

On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
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0answers
43 views

Does 1/4 eventually repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's ...
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0answers
116 views

Do all rational numbers repeat in Fibonacci coding?

In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's ...
0
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1answer
47 views

Need help calculating probability…

First time here, so I hope you'll not get too frustrated if I make any etiquette mistakes for this forum. So here's my question. I know there are snow day calculators out there, but I'm trying to ...
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0answers
24 views

Find length of sequence with fibonacci principle

Let assume we have sequence: $${l_{n + 1}} = {l_n} + {l_{n - 1}} $$ $$\begin{array}{l}{l_0} = 0\\{l_1} = 01\\{l_2} = 010\end{array}$$ Our goal to get $|{l_n}|$ (length). For ${l_0}$ it's 0, for ...
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2answers
42 views

Proof and geometrical significance of $F(n+1)^2-F(n) \cdot F(n+2)$?

My son notes that for Fibonacci numbers $F_n$, $$ (F_{n+1})^2-F_n \cdot F_{n+2} =(-1)^n $$ I assume that this is true. Update: I see that the proof is already here: Prove the given property of the ...
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0answers
41 views

Is the following statement true: $F_n\leq p\leq F_{n+2}$

Let be $n\in\{2,3,\ldots\}$ Is the following statement true: There exists a prime number so that $F_n\leq p\leq F_{n+2}$ while $F_n$ is a Fibonacci number.
3
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1answer
60 views

Fibonacci sequence digits

We define the Fibonacci sequence by $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0$ and $F_1=1$. How to compute the last $30$ digits of $F_{2^{200}}$ for instance? can we use Python?how?
2
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0answers
76 views

Fibonacci Numbers and the Harmonic Series

$$\sum_{k=1}^{n} \frac{1}{k}=H_n=\frac{p_n}{q_n}$$ Where $p_n,q_n$ are coprime intergers. The first few values for $p_n+q_n$ are $2,5,7,37,197,69,504,1041,9649$. When are $p_n+q_n$ Fibonacci ...
4
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1answer
96 views

Some heuristics about the Pisano Period, primes and Fibonacci primes. What reasons are behind them?

I started to read about the Pisano Period, $\pi(n)$, applied to the classic Fibonacci sequence and made some simple tests looking for possible properties of the sequence. I have observed the following ...
0
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1answer
31 views

Is there some function which return probability to select prime number from $n$ first Fibonacci numbers.

So my question is: is there function return probability to select prime number from $n$ first Fibonacci numbers. So maybe it realize with $\pi(n)$ function?
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2answers
87 views

Domino tiling extended in N dimensions.

The standard domino tiling problem, is the number of ways to tile a board of size 2xn by dominos of size 2x1. The answer directly follows a recursion, the same as the Fibonacci series. If I extend ...
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1answer
34 views

How is this identity for Fibonacci numbers called?

In the course of proving another identity, I've found that $$F_n \equiv F_kF_{n-k+1} + F_{k-1}F_{n-k}$$ …for all corresponding n and ...
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0answers
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Are all totient values of Fibonacci Numbers distinct?

This question was inspired while I was seeing how certain recurrence relations would behave when I applied Multiplative Functions. Let $F_{n}$ be a sequence for which $F_{1}=1,F_{2}=1$, and ...
2
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1answer
40 views

Is it rational or not?

I have two interesting question : Is this number rational or not: $$0.F_{1}F_{2}F_{3}...$$, where $F_{i}$ - Fibonacci number. And is this number rational or not: $$0.p_{1}p_{2}...$$
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2answers
72 views

Find $\mathrm{lcm}(F_{m},F_{n},F_{k})$.

So consider three Fibonacci numbers. My question is: Find $\mathrm{lcm}(F_{m},F_{n},F_{k})$, where lcm is least common multiple.
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0answers
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Is the “Fibonacci square tiling” of Fibonacci-sided rectangles always optimal?

Is an optimal square tiling of a rectangle with side lengths of successive Fibonacci numbers always the sequence of Fibonacci numbers, as in the picture below?
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1answer
61 views

Convergence of Fibonacci quotients

Let $F_n=F_{n-1}+F_{n-2},~ F_0=0,~F_1=1~$ be the Fibonacci numbers. Then it is well known that $\lim_n \frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$. However, many textbooks proved the above by using ...
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1answer
47 views

Fibonacci identity $F_{n+1}^2 - (F_{n+1}F_n) - F_n^2 = (-1)^n$

I am trying to prove Let $F_n$ be the $n$th Fibonacci number. Then $F_{n+1}^2 - F_{n+1}F_n - F_n^2 = (-1)^n$ I am not sure where to start with this.
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2answers
26 views

If $y_n = 2x_n-1$, show that $y_{n+1} = y_n + y_{n-1} + 1$

If $y_n = 2x_n-1$, how do you show $y_{n+1} = y_n + y_{n-1} + 1$ with $y_0 = 1$ and $y_1 = 1$? Would you start with $y_{n+1} = y_n + y_{n-1} + 1$, find a formula for $y_n$ and then compare it with ...
0
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1answer
49 views

How do you show $s_n = \frac{x_{n+1}}{x_n}$ where $(x_n)$ is the Fibonacci sequence?

Let $(s_n)$ denote the sequence satisfying: $s_{n+1} = 1 + \frac{1}{s_n}$ with $s_0 = 1$. Let $(x_n)$ denote the Fibonacci sequence and $x_n = \frac{5 + \sqrt{5}}{10}(\frac{1 + \sqrt{5}}{2})^n + ...
1
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1answer
65 views

Limit as $n \to \infty$ of $\frac{x_{n+1}}{x_n}$ in the Fibonacci sequence

Given that $x_n = \frac{\sqrt{5}+5}{10} (\frac{1 + \sqrt{5}}{2})^n + \frac{5 - \sqrt{5}}{10}(\frac{1 - \sqrt{5}}{2})^n$ How do you show that $\lim_{n \to \infty} \frac{x_{n+1}}{x_n}$ is the ...
0
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1answer
57 views

formula for the nth term of this sequence?

How do you find a formula for the nth term of this sequence? given that $x_n$$_+$$_1$ = $x_n$ + $x_n$$_-$$_1$ (Fibonacci sequence) and $x_0 = 1$ and $x_1 = 1$. Do i complete the square on $x^2 - x - ...
1
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2answers
38 views

Finding the limit of the following series, wherever it is convergent

Let $ f(x) = \sum_{n=1}^{\infty}a_nx^n $ where $a_n$ is the nth Fibonacci number. Find the values for which the series is convergent, and find $f(x)$ for those values. (i.e - find the limit when ...
5
votes
1answer
63 views

Limit of a specific sequence involving Fibonacci numbers.

Let, $\left\{F_n\right\}_{n=1}^\infty$ be the Fibonacci sequence, i.e, $F_1=1, F_2=1~\&~ F_{n+2}=F_{n+1}+F_n~\forall ~n \in \mathbb{Z}_+$ Let, $P_1=0, P_2=1$. Divide the line segment ...
1
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2answers
47 views

Proof concerning Fibonacci and recursively defined sequences

The series $(a_n)_{n\in\mathbb{N}}$ is given through $$a_1=1,\quad a_2=\frac{1}{2},\quad a_{n+2}=a_na_{n+1} \quad\text{ for } n\geq1.$$ I want to show that $$a_n = 2^{-f_{n-1}}$$ whereas ...