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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4
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1answer
198 views

Relation between Fibonacci number and the golden section

We denote the $n$th term of Fibonacci number with $F_n$. Assume that $\alpha=\frac{1+\sqrt{5}}{2}$. With simulation, I found the following relation between Fibonacci number and the golden section $$ ...
0
votes
1answer
26 views

How do I interpret following equations on fibonacii numbers?

I went through an online tutorial (http://codeforces.com/blog/entry/14385) on finding n-th fibonacci number which explains a method as, You are standing at position n in Ox axis. In a step, ...
10
votes
4answers
3k views

Fibonacci sequence divisible by 7?

Make and prove a conjecture about when the Fibonacci sequence, $F_n$, is divisible by $7$. I've realized it's when $n$ is a multiple of $8$. I just don't know how to go about proving it.
3
votes
2answers
95 views

What is the limit of the sequence: $n$-th root of the $n$-th Fibonacci number?

My computer can not calculate numbers large enough for this. If you take the $n$-th Fibonacci number $F_n$ and raise it to the $1/n$-th term, where does the sequence $F_n^{1/n}$ tend to? Examples: ...
14
votes
4answers
5k views

Fibonacci modular results

Can any one give a generalization of the following properties in a single proof? I have checked the results, which I have given below by trial and error method. I am looking for a general proof, which ...
2
votes
2answers
53 views

Fibonacci using Matrix Representation.

Fibonacci using matrix representation is of the form : Fibonacci Matrix. This claims to be of O(log n).However, isn't computing matrix multiplication of order ...
4
votes
1answer
96 views

What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

Mathematica is able to evaluate the summation $\sum_{i=1}^{n}\frac{F_i}{i}$ in terms of the Lerch transcendent. It is natural to consider whether or not this summation can be expressed in a more ...
0
votes
1answer
39 views

A sequence like fibonacci sequance that has got the shortest formula. [closed]

I want a squance like this: $X_1=a$,$X_2=b$,$X_n=X_{n-2}+X_{n-1}$ But I want one that has the shortest formula for example I found the lucas numbers formula.it was shorter than the fibonacci numbers ...
9
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1answer
135 views

Is $n^7 - 77$ ever a Fibonacci number?

As the question title suggests, is $n^7 - 77$ ever a Fibonacci number, where $n$ is a integer?
0
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1answer
42 views

Fibonacci numbers in nature(Give examples)

I heared a lot of things about fibonacci numbers in nature my friend says that there are about $300$ ways to find fibonacci numbers in nature but when I searched I can see just the sun flower and the ...
8
votes
1answer
234 views

Is the Fibonacci constant $0.11235813213455…$ a normal number?

Recall that a normal decimal number is an irrational number $\alpha \in \mathbb{R}$ such that each digit 0-9 appears with average frequency tending toward $\frac{1}{10}$, each pair of digits 00-99 ...
1
vote
3answers
71 views

Product of the first $n$ Fibonacci numbers is a perfect square

Suppose that $F_{n+2}=F_n+F_{n+1}$ and $F_1=F_2=1$. Can the number $P_n=F_1\cdots F_n$ be a perfect square if $n\ge 3$?
1
vote
1answer
23 views

Proving i-th Fibonacci number by induction, can an inductive step be used for two sequential values?

I am working through the beginning of Introduction to Algorithms, and came across the problem Prove by induction that the $i$-th Fibonacci number satisfies the equality $$ F_{i} = \frac{\phi^{...
2
votes
0answers
54 views

Fibonacci summation

Can anyone help me to prove the following relation. $$\sum_{k=1}^{\infty} \frac{F_{2k}H^{(2)}_{k-1}}{k^2\binom{2k}{k}}=\frac{2\pi^4}{375\sqrt{5}}$$ I was studying recently about Fibonacci and ...
5
votes
3answers
96 views

Each prime $p$ not $2$ or $5$ divides $F_{p-1}$ or $F_{p+1}$, where $(F_n)$ is the Fibonacci sequence with $F_1=F_2=1$

Let $\{F_n\}$ - Fibonacci sequence: $F_1=F_2=1, F_{n+1}=F_n+F_{n-1}, n\ge2$ and $p -$ prime number, $p\not =2, p \not=5$. Prove that $p|F_{p-1}$ or $p|F_{p+1}$ My work so far. I used formula $$F_p^...
17
votes
3answers
232 views

Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: $$F(z)=\left(\phi^...
5
votes
1answer
66 views

Sum of nth powers of Fibonacci numbers

Is a closed form for $$\sum_{i=1}^n{F_i^k}$$ (where $F_i$ is the $i^{th}$ Fibonacci number and $k$ is constant) known?
5
votes
1answer
147 views

Number of zeros in Fibonacci sequences mod $p$

We know that Fibonacci sequences are periodic in mod $m$. For example, for $p\equiv \pm 1 \pmod 5$ and $\pm 2 \pmod 5$ the periods for Fibonacci sequences modulo $p$ divide $p-1$ and $2p+2$ ...
1
vote
2answers
46 views

Prove the n-th Fibonacci number is less than $2^n$ for all n greater than zero using strong induction

I need to prove the n-th Fibonacci number is less than $2^n$ for all $n \geq 0$ using strong induction. I have been exposed to the idea that strong induction differs from weak induction in that the ...
3
votes
5answers
305 views

Summation of Fibonacci numbers $F_n$ with $n$ odd vs. even

Compare the summation below: $$\begin{align} \smash[b]{\sum_{i=1}^n F_{2i-1}}&=F_1+F_3+F_5+\cdots+F_{2n-1}\\ &=1+2+5+\cdots+F_{2n-1}\\ &=F_{2n}\\ \end{align} $$ with this one: $$\begin{...
3
votes
3answers
81 views

Limit of fibonacci sequence

Let $f_n$ be the $n$th Fibonacci number. Find constants $a$ and $b$ such that $$\lim_{n\to\infty} \frac{f_n}{a\cdot b^n} = 1$$ I'm somewhat confused on how to approach this problem. I know the ...
0
votes
1answer
24 views

Linear combination of sequence related to Fibonacci

Let $a= \frac{1+\sqrt{5}}{2}$ and $b= \frac{1-\sqrt{5}}{2}$. Prove that for all $c,d \in \mathbb{N}$ that $g_n = ca^n + db^n$ satisfies $g_{n+2} = g_{n+1} + g_{n}$ for all $n\in \mathbb{N}$. I'm ...
1
vote
1answer
80 views

Find all positive integers $m>1$ such that any sum of $m$ consecutive Fibonacci numbers is divisible by $m$.

Let $\{u_n\} -$ Fibonacci sequence: $u_1=u_2=1, u_{n+1}=u_n+u_{n-1}, n\ge2$. Find all positive integers $m>1$ such that any sum of $m$ consecutive Fibonacci numbers is divisible by $m$. My work ...
1
vote
0answers
27 views

Lucas recurrence relation

Lucas recurrence relation is defined as: $V_n = PV_{n-1} – QV_{n-2}$; for $V_0 = 2; V_1 = P$ Here $P$ is positive integer and $Q = {-1, 1}$ or may be $+1$ or $-1$ A Fibonacci Pseudo prime with ...
3
votes
1answer
640 views

Dominos ($2 \times 1$ on $2 \times n$ and on $3 \times 2n$)

How many ways are there to tile dominos (with size $2 × 1$) on a grid of $2 × n$? How about on a grid of $3 × 2n$?
2
votes
1answer
30 views

Greatest number in fibonacci sequence with property: sum of digits=index in fibonacci sequence

I came across very interesting sequence based on fibonacci sequence. From fibonacci numbers we choose only elements with digit sum=index in fibonacci sequence. It is very interesting that we most ...
1
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1answer
49 views

derivation of fibonacci log(n) time sequence

I was trying to derive following equation to compute the nth fibonacci number in O(log(n)) time. F(2n) = (2*F(n-1) + F(n)) * F(n) which i found on wiki form the ...
3
votes
1answer
48 views

Relationship between decimal length and Fibonacci number

There are 6 single digit Fibonacci numbers. For all other number of digits in the decimal system, there are either 4 or 5 Fibonacci numbers. For example, between 10000 and 99999 there are 5: 10946 ...
6
votes
1answer
132 views

Prove or disprove that ${F_{n}^2} + 43$ is always a composite

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is ...
4
votes
2answers
74 views

Proving that $f_2+f_4+\cdots+f_{2n}=f_{2n+1}-1$ for Fibonacci numbers by induction

Given: $f_1 = f_2 = 1$ and for $n \in\mathbb{N}$, $f_{n+2} =f_{n+1} + f_n$. Prove that $f_2 + f_4 + \dots + f_{2n} = f_{2n+1}- 1$. Would you start with setting $f_2 + f_4 + \dots + f_{2n}= ...
0
votes
0answers
26 views

How to prove that three different modulo 9 equations results the same sequence?

First let index sequence $ℕ_0=(0,1,2,…)$ and $n∈\mathbb{N}_{0}$. Then let: $$S_a = (-1)^n(a+bn) \text{ mod 9 } \text{ where } a = 1\text{, } b = -3$$ $$S_b = 2^n \text{ mod 9 }$$ $$S_c = F_{a+bn} \...
2
votes
1answer
152 views

Show that $5^n$ divides $F_{5^n}$.

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 $5^n$ divides $F_{5^n}$.
1
vote
1answer
52 views

Reverse and forward doubling identity in Fibonacci sequence $\text{mod 9}$

Fibonacci sequence ($\mathbb{F}$) has a repeating cycle known as Pisano number $\pi\text{(x)}$ , when $mod \text{ x}$ is applied upon the sequence. Length of the cycles can be found from: http://oeis....
7
votes
1answer
672 views

Product of Fibonacci numbers

I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers. Does there exist a tight approximation for these kind of things?
0
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0answers
38 views

Relating fibonnaci sequence, lucas numbers and golden ratio to make a research question?

I am planning to write a high school level maths essay of approximately 4000 words. I do find Fibonacci sequence, Lucas numbers and Golden ratio amazing and want to research further on them, the thing ...
5
votes
5answers
15k views

Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$

I can't seem to solve this problem. It is: The Fibonacci numbers $F(0), F(1), F(2),\dots $ are defined as follows: \begin{align} F(0) &::= 0 \\ F(1) &::= 1 \\ F(n) &::= F(n-1) + F(...
1
vote
3answers
47 views

For a sequence $\{f_n\}_{n\geq 1}$ defined as…

For a sequence $\{f_n\}_{n\geq 1}$ defined as: $f_n= \frac{F_n}{F_{n+1}}$, where $F_n$ is the $n^{th}$ term in the Fibonacci series, starting with a $1$ rather than $0$. I wish to find: $\lim_{n \to \...
1
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1answer
54 views

Sum of $resiprocals$ of the $Fibonacci$ $series$

Well I was having a doubt on the infinite sum of the reciprocals of the $Fibonacci$ $series$. That is: $S=1+1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+....$ Assuming that the $series$ starts with $1$ ...
0
votes
1answer
47 views

Strong Induction for Fibonacci number related identity $f_{n-m} = f_{m}f_{n+1} + f_{m-1} f_n$ [closed]

Let $f_n$ be the $n^{th}$ Fibonacci number. Let $m$ be a fixed strictly positive integer. Prove by strong induction that for all $n\ge 0$, $$f_{n+m} = f_{m}f_{n+1} + f_{m-1} f_n$$ edit: $f_{n+m} = ...
0
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0answers
21 views

Constant value of a sequence

I am writing a article about sequence number. Is it acceptable to write two words "constant value" for limitation of two successive terms of a sequence. I mean, assume Fibonacci number, I used "...
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0answers
34 views

Prove by induction that $ F_{2n}=F_nL_n $

In the following exercise from George E. Andrews' Number Theory, we are given that $F_n$ and $L_n$ represent the $nth$ Fibonacci and Lucas numbers respectfully, and we need to prove by induction (i.e. ...
22
votes
1answer
519 views

Fibonacci-related sum

Related to this question Find a solution for f(1/x)+f(1+x)=x, what is this sum: $$\sum_{n=1}^{\infty}(-1)^n\left(\frac{F_n}{F_{n+1}}-\frac1{\phi}\right)$$ where $F_n$ is the $n$th Fibonacci number and ...
0
votes
0answers
48 views

Fibonacci sequence matrix

Let us recall that the famous Fibonacci sequence: $0,1,1,2,3,5,8,13,21,\dots$ is defined as follows: we put $\phi_0 = 0, \phi_1 = 1$ and define $\phi_{n+2} = \phi_{n+1} + \phi_n$. We want to find a ...
3
votes
1answer
59 views

Why Fibonacci(prime-1) or Fibonacci(prime+1) is divisible by that prime?

Why Fibonacci(prime-1) or Fibonacci(prime+1) is divisible by that prime and Fibonacci(nonprime-1) or Fibonacci(nonprime+1) is not divisible by that nonprime? Is there any elegant proof of that?
22
votes
2answers
12k views

Prove that two any consecutive terms of Fibonacci sequence are relatively prime

Prove that two any consecutive terms of Fibonacci sequence are relatively prime My attempt: We have $f_1 = 1, f_2 = 1, f_3 = 2...$. So obviously $\gcd(f1, f2) = 1$. Suppose that $\gcd(f_n, f_{n+1}...
1
vote
1answer
56 views

What is the Fibonacci-like sequence called where one sums the last 3 numbers

The Fibonacci-sequence is defined like. $F_{x+1} = F_{x} + F_{x-1}; F_0 = 0, F_1=1, x \in {\Bbb N}$ Is there a special name for this sequence: $F_{x+1} = F_{x} + F_{x-1} + F_{x-2}$ ? Which?
34
votes
5answers
13k views

Checking if a number is a Fibonacci or not?

The standard way (other than generating up to $N$) is to check if $(5N^2 + 4)$ or $(5N^2 - 4)$ is a perfect square. What is the mathematical logic behind this? Also, is there any other way for ...
1
vote
1answer
78 views

What is the best way to inscribe a golden rectangle into a pentagon? Do more golden ratios emerge?

Below I drew a golden rectangle in a pentagon in Adobe Illustrator. What would be the best way to inscribe a golden rectangle into a pentagon as shown in the figure below in a mathematical manner? ...
0
votes
2answers
40 views

What Does a Subscript Do to a Number? [closed]

So I had a math question that had a formula for that said Tn= arn-1 Where a is the first sequence and r is the common ratio. For example, in the sequence 10,40,160,640,..., a=10, and r=40/10=160/40=...
0
votes
3answers
133 views

Help on require answer $\int_0^1\frac{2}{[1+x+(-\phi)^{-n}(1-x)]^2}dx=\frac{1}{-{\phi}F_n+F_{n+1}+1}$

$\phi=\frac{1+\sqrt5}{2}$ $F_0=0$, $F_1=1$ ;$F_{n+1}=F_n+F_{n-1}$ ; Fibonacci numbers (0,1,1,2,3,5,...) Show that, $$\int_0^1\frac{2}{[1+x+(-\phi)^{-n}(1-x)]^2}dx=\frac{1}{-{\phi}F_n+F_{n+1}+1}$$ ...