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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How to find the root value of a fibonacci sequence from two consecutive values?

I'm stuck on a question that was given to me on a list of problems. It's to do with the fibonacci sequence such that if I have the a=f(n) and b=f(n+1) of a sequence that is a fibonacci sequence, find ...
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1answer
201 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 $$ ...
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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, ...
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2answers
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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: ...
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2answers
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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 ...
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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 ...
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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 ...
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1answer
137 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?
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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 $n\ge 3$?
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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 ...
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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^{...
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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 ...
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3answers
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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^...
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1answer
235 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 ...
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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?
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2answers
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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 ...
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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 ...
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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 ...
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0answers
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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 ...
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1answer
89 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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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} \...
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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}= ...
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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 $F_{5^n}$.
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1answer
53 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....
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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?
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0answers
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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 ...
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3answers
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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 \...
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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 $1$ ...
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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} = ...
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0answers
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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. ...
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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
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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?
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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?
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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^...
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2answers
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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=...
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0answers
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Number of ways to express a Fibonacci number as the sum of N other Fibonacci Numbers.

How do I find the number of ways to express a Fibonacci number as the sum of N other Fibonacci Numbers? There can be repetitions. Consecutive Fibs are allowed.
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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? ...
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1answer
48 views

Induction proof for Fibonacci sum different notation

This question was asked but using sum notation and I am trying to relate it to what I am doing. I am trying to prove by induction that for the Fibonacci series, $a_1+a_2+...+a_n=a_{n+2}-1$ is true. $...
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3answers
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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, $$\int_0^1\frac{2}{[1+x+(-\phi)^{-n}(1-x)]^2}dx=\frac{1}{-{\phi}F_n+F_{n+1}+1}$$ ...
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2answers
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Proof of $1^3+1^3+2^3+3^3+5^3+\cdots +F_n^3=\frac{F_nF_{n+1}^2+(-1)^{n+1}[F_{n-1}+(-1)^{n+1}]}{2}$

Fibonacci series $F_0=0$, $F_1=1$; $F_{n+1}=F_n+F_{n-1}$ This is a well known identity $1^2+1^2+2^2+3^2+5^2+\cdots +F_n^2=F_nF_{n+1}$ I was curious and look every websites for a closed form of $1^...
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1answer
73 views

Golden Ratio & Fibonacci - Charles de Gaulle 13-unit two-beamed cross problem.

Here is the question: The two-beamed cross, made popular by Charles de Gaulle, is formed from 13 unit squares as shown below. A straight line $BC$ drawn through point $A$ divides the cross in such ...
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6answers
354 views

What is the sum of all the Fibonacci numbers from 1 to infinity.

Today I believe I had found the sum of all the Fibonacci numbers are from $1$ to infinity, meaning I had found $F$ for the equation $F = 1+1+2+3+5+8+13+21+\cdots$ I believe the answer is $-3$, however,...
3
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2answers
78 views

Fibonacci Pairs

Find all positive integer solutions to $y^2 - xy - x^2 = 1$ and $y^2 - xy - x^2 = -1$ I have written a C++ program to yield some solution for large constants. I must make conjectures based on the ...
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1answer
35 views

Doubt in a property of the Fibonacci Series.

I came across this question in a book where they asked me to prove that there are exactly four terms such that $F_{F_n}= F_n$. Well, I think that this is false and that there are exactly three. I have ...
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1answer
22 views

Exponential generating function and fibonnaci

$F_n$ is the $n$th Fibonnaci number. $$g(x) = \sum^\infty_{n=0}F_n \frac{x^n}{n!}$$ Prove that $$g''(x)=g'(x)+g(x)$$ I've never dealt with derivatives in the above form so I am not exactly sure ...
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0answers
39 views

Proofing that the Lucas numbers come closer to the Phi rounded numbers then the Fibonacci numbers.

Morning everyone, Bit of background, I'm a mid level programmer with very limited mathematics skills. As part of an assessment for a new role I've been asked to complete a technical task which ...