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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Squares Containing digits from Fibonacci Numbers

This is a problem that I thought of myself. Currently, the digits of $F_1$ through $F_n$ are written on the chalkboard in order, where ${F_n}$ is the Fibonacci Sequence and $n \ge 5$ We shall ...
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1answer
25 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 ...
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1answer
39 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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23 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; $ $F_{(1)-1} = F^2_{(1)} + F^2_{(1)-1} ->$ $ ...
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2answers
28 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 ...
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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 ...
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1answer
303 views

How to prove this series about Fibonacci number?

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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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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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: ...
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1answer
21 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, ...
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2answers
158 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 ...
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1answer
60 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
38 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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94 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 ...
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1answer
41 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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22 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
41 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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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.
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1answer
54 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?
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65 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 ...
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1answer
77 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 ...
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1answer
30 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
80 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
33 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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122 views

Are all totients 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 ...
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1answer
37 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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53 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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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
56 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
44 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 ...
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1answer
47 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 + ...
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1answer
62 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 ...
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1answer
44 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 - ...
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2answers
37 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 ...
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1answer
58 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 ...
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44 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 ...
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2answers
65 views

Seeking a combinatorial proof $F_{mn}$ always a multiple of $F_m$

I would appreciate if somebody could help me with the following problem Q: Let $F_n$ the sequence of Fibonacci numbers, given by $F_1 = 1, F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$ ...
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1answer
77 views

Integral representation for Fibonacci's numbers

We know that, for example, the Gamma function is a perfect integral representation for the factorial $n!$ for a natural number $n$. $$\Gamma[n] = \int_0^{+\infty} t^{n-1}e^{-t}\text{d}t = (n-1)!$$ ...
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130 views

Is there a proven way to calculate the entry point(first occurence) of a factor m, in the Fibonacci sequence?

I saw a comment at the OEIS website for the sequence of entry points, of Fibonacci factors. https://oeis.org/A001177 It referenced a paper by Mark Renault in 1996, with the quote from OEIS: ...
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51 views

Do I need induction here?

I am asked to prove, by using induction that $$\sum\limits_{i=1}^n F(2i-1) = F(2n)$$ for all real numbers n where the function F(i) gives the i:th fibonacci number. The series stars off with $F(0) ...
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4answers
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1answer
50 views

Convergence of Series Whose Terms are Defined Recursively

My recursively defined sequence $(a_n)_{n\in\mathbb{N}}$ is given trough $$a_1 = 1, \quad a_2=\frac{1}{2}\quad a_{n+2}=a_{n}a_{n+1}\quad \text{for } n\geq1$$ and I have to show that the series ...
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2answers
22 views

Closed form expression for zero of recurrence relation

Given the recurrence $d(i+1)=xFib(2i+1)-nFib(2i)$, where $Fib$ denotes the Fibonacci sequence (i.e. $Fib(0)=0, Fib(1)=1, Fib(2)=1, Fib(3)=2$, etc) and $n$ and $x$ are arbitrary integers, is it ...
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45 views

How to prove this sequence is null?

I am working on the fibonacci numbers series using the ratio. To prove convergence I want to show that the sequence of the series is going to 0. And then according to the Leibniz criterion the series ...
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0answers
37 views

Fibonacci numbers and binomial theroem [duplicate]

So I am trying to prove $$\sum_{i=0}^n{nCi×F_i} = F_{2n}$$ Such that $$nCi = \frac {n!}{i!×(n-i)!}$$ And $F_i$ is the ith value of the fibonacci sequence such that $F_0 = 1$ and $F_1 = 1$ I have ...
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1answer
51 views

Proof: Fibonacci Sequence (2 parts)

Part a) Prove or Disprove: There are only finitely many even Fibonacci numbers. I think I want to disprove this, as I know that every 3rd Fibonacci number is even, and thus there will be infinitely ...
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29 views

Divisibility of Fibonacci Sequence mod prime

I have to solve the following problem and I have a few questions: Consider the Fibonacci sequence defined as $F_n:=2F_{n-1}+F_{n-2}$ with $F_0=1$ and $F_1=1$. Now, I need to prove that for any odd ...
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3answers
71 views

Cesaro identity for Fibonacci numbers

I am stuck with the identity $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$ which happens to be formula 80. I am using induction, but so far without too much result. $$ \sum_{k=1}^{n+1} ...
4
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1answer
327 views

Given Two Fibonacci numbers, predicting the median Fibonacci number

Wolfram Alpha gives the $100$th fibonacci number to be $354224848179261915075$ and the $104$th fibonacci number to be $2427893228399975082453$. Just from this, can we deduce what the $102$th fibonacci ...