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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5answers
290 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{...
17
votes
2answers
208 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^...
3
votes
3answers
74 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
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1answer
23 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
69 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
24 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
594 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 $3 × 2n$?
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1answer
27 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
46 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
47 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
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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
70 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
144 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
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1answer
48 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
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1answer
666 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
35 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
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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(...
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3answers
45 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 \...
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1answer
53 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
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1answer
43 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
20 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
33 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
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1answer
510 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
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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
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1answer
56 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
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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
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1answer
77 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
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2answers
39 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
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3answers
131 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, $$\int_0^1\frac{2}{[1+x+(-\phi)^{-n}(1-x)]^2}dx=\frac{1}{-{\phi}F_n+F_{n+1}+1}$$ ...
0
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0answers
249 views

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
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. $...
2
votes
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 ...
8
votes
4answers
807 views

Sum of cubes of first n fibonacci numbers

Let $\{f_k\}$ be the sequence of fibonacci numbers. It is well-known that $\sum_{k=1}^n f_k=f_{n+2}-1$ and $\sum_{k=1}^n f_k^2=f_n f_{n+1}$ . Is there a formula for $\sum_{k=1}^n f_k^3$ ?
3
votes
2answers
69 views

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^...
0
votes
6answers
340 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
votes
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 ...
0
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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 ...
0
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0answers
33 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 ...
3
votes
1answer
89 views

Is there a number besides $\phi$ that either squared or added one gives the same answer?

Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very ...
1
vote
2answers
45 views

Prove equality for Fibonacci sequence [duplicate]

I have to show that $F_{n+k} = F_{k}F_{n+1} + F_{k-1}F_{n}$, where $F_{n}$ is nth Fibonacci element. I was trying with mathematical induction applied to n and saying k is constant. step for $n=1$ ...
0
votes
4answers
42 views

Prove that the given property of the fibonacci number directly from the definition

$F(n)= 3F(n-3)+2F(n-4)$ for $n \ge 5$. I just don't understand the whole process of this, my instructor has a rather weird way of explaining and I couldn't understand. can someone help me ...
2
votes
0answers
38 views

An identity for $q-$Fibonacci numbers if $q$ is a root of unity.

In his proof of the Rogers-Ramanujan identities I. Schur introduced two $q-$analogues of the Fibonacci numbers ${F_n}({q})$ and ${G_n}({q})$, which satisfy ${F_0}({q})=0$, ${F_1}({q})=1$ and $${F_n}...
0
votes
3answers
97 views

Prove $\phi^n = \phi F_n + \text{(another Fibonacci number)}$ using mathematical induction.

I need to prove the following equation using mathematical induction and using the phi values if necessary. $\phi^n = \phi F_n + \text{(another Fibonacci number)}$ In this proof, it is kind of hard ...
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0answers
43 views

Mathematical proof by induction. [duplicate]

How to prove the following using mathematical proof by induction? $\phi^n = \phi\times F_n + F_{n-1}$ $\phi = 1 + \sqrt 5 /2$ Fn is the Fibonacci number. I tried solving this using induction but ...
2
votes
2answers
49 views

Restrictions for rule of general sequence: $T_{n+2}=T_{n} + T_{n+1}$

This is a rule which gives a sequence where $11$ times the seventh term ($11 t_7$) is equal to the sum of the first $10$ terms ($s_{10}$), where the first two starting numbers can be chosen. Do any ...
0
votes
1answer
38 views

Fibonacci numbers and proving using mathematical induction

I need to prove the rule below using Mathematical proof by induction but actually I'm stuck in the middle of the proving. $$F_n^2 - F_{n-1}F_{n+1} = (-1)^{n+1} {~\rm for~} n>1$$ If someone can ...