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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4answers
39 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 $F = 1+1+2+3+5+8+13+21\ldots$ I believe the answer is $-3$, however, I ...
3
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1answer
42 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
23 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 ...
1
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1answer
19 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
26 views

PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle?

This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ...
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0answers
25 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
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1answer
88 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
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2answers
41 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
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4answers
37 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 ...
1
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0answers
31 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 ...
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3answers
83 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
40 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 ...
1
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2answers
44 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 ...
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1answer
36 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 ...
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0answers
30 views

Proving the GCD property of the Fibonacci numbers [duplicate]

How to prove the following GCD property of the Fibonacci number sequence using mathematical proof by induction? gcd (Fn , Fm) = F (gcd(n , m)) Help is much appreciated. Thank you in advance.
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1answer
56 views

Show that $\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10}$

Show that$$\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10},$$ where $F_n$ is a Fibonacci number and $L_n$ is a Lucas number.$^1$ Motivation: For example, when ...
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0answers
39 views

Proofs with Fibonacci and Lucas numbers via induction

How would I go about proving the following sequence using induction on $k$? $2F_{2n+k} = F_{n+k}L_n + F_nL_{n+k}$ I know I have to show that it's true for $k = 1$, but I can't even seem to be able ...
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0answers
26 views

Fibonacci Sequence and Time taken

Consider the following (incomplete) java code, which calculates the Fibonacci numbers ...
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2answers
144 views

Deducing the closed form for pentagonal numbers

Consider the sequence: $0,1,5,12,22,35,51,70,92,117,145,176,\ldots$ Find both a recurrence and a closed form for this sequence. I've done some research and found out that the majority ...
40
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5answers
973 views

Is ${F_{n}}^2 - 28$ always a composite number?

The problem is as follows: Prove or disprove that if ${F_{n}}$ is $n$-th Fibonacci number, and $n>5$, than $${F_{n}}^2 - 28$$ cannot be a prime. I came across this problem accidentally ...
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0answers
17 views

Pisano Period - Fibonacci

I have to construct an algorithm which will return a fibonacci number mod another integer. I know that i have to implement the Pisano period. I know how to receive it, but the problem comes when I ...
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3answers
33 views

Prove that the $c_{n}$ in $\frac{1}{1-z-z^{2}}=\displaystyle\sum_{n=0}^{\infty}c_{n}z^{z}$ satisfy a Fibonacci-like recurrence relation

I need to prove that the coefficients $c_{n}$ of the expansion $\frac{1}{1-z-z^{2}}\sum_{n=0}^{\infty}c_{n}z^{n}$ satisfy the recurrence relation $c_{0}=c_{1}=1$, $c_{n}=c_{n-1}+c_{n-2}$, by ...
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1answer
44 views

Alternative “Fibonacci” sequences and ratio convergence

So the well known Fibonacci sequence is $$ F=\{1,1,2,3,5,8,13,21,\ldots\} $$ where $f_1=f_2=1$ and $f_k=f_{k-1}+f_{k-2}$ for $k>2$. The ratio of $f_k:f_{k-1}$ approaches the Golden Ratio the ...
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1answer
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Another way to go about proving the limit of Fibonacci's sequence quotient.

It is not difficult to inductively prove that $$\eqalign{ & \phi = \phi + 0 \cr & {\phi ^2} = \phi + 1 \cr & {\phi ^3} = 2\phi + 1 \cr & {\phi ^4} = 3\phi + 2 ...
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0answers
55 views

Golden Ratio To find Fibonacci Number

I have to find the Xth Fibonacci Number i.e F(X)%1000000007For Example If i have to find X(350) = 672262724 Code Now i am ...
9
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1answer
145 views

Number of ways to write $n$ as sum of odd or even number of Fibonacci numbers

In our discrete mathematics exercises I came of with the question: Prove that the coefficients of $\prod_{n\geq2}{(1-x^{F_n})}=1-x-x^2+x^4+x^7+\dots$ can only be $-1,1$ or $0$, where $F_n$ ...
3
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1answer
78 views

Closed form for Fibonacci numbers

We know the closed form for Fibonacci number as $F_n=\frac{1}{\sqrt5}\left[\left(1+\frac{\sqrt5}{2}\right)^n−\left(1−\frac{\sqrt5}{2}\right)^n\right]$ But while finding $F_n \pmod{99991}$ the closed ...
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3answers
56 views

Recurrence Relation of a series

I know this would seem lame, but I need to ask this. I was trying to solve some recurrence based problems and I came across this series. $1,2,4,7,12,\dots$ Question was: To find the recurrence ...
0
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1answer
62 views

Fibonacci polynomials

The Fibonacci polynomials are defined by the recurrence relation: $$ F_{n+1}(x)=xF_{n}(x)+F_{n-1}(x)\, . $$ with $F_1(x)=1$ and $F_2(x)=x$. How can I prove: $$ F_n(x)=\sum_{j=0}^{\lfloor \frac{n-1}{2} ...
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2answers
55 views

How can I prove by induction that this is a closed form of the Fibonacci sequence? [duplicate]

How can I prove by induction that this is a closed form of the Fibonacci sequence? $$F_n=\frac1{\sqrt5}\left(\frac{1+\sqrt5}2\right)^{n+1}-\frac1{\sqrt5}\left(\frac{1-\sqrt5}2\right)^{n+1}$$ I've ...
1
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1answer
123 views

Formula for fibonacci(a+b).

Is there any general formula for fibonacci(A+B)? I have tried to derive it , and got following results. $$\begin{align} &fib(a+1)=1*fib(a)+fib(a-1)\\ &fib(a+2)=2*fib(a)+fib(a-1)\\ ...
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2answers
295 views

Magic Squares with Lucas and Fibonacci Numbers

I am quite curious about can we construct magic squares using only Lucas and Fibonacci numbers(of course not repeating them? If yes, how can we construct them? And if not , what is the proof?
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5answers
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Prove this formula for the Fibonacci Sequence

This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$. Show that ...
0
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1answer
60 views

Partition function and Fibonacci n-th number upperbound

i need to proof this upperbound: "Call $p(n)$ the number of partitions of n (integer) and $F_n$ the $n-th$ Fibonacci number. Show that $p(n)\leq p(n-1) + p(n-2)$ and that $p(n)\leq F_n$ for $n\geq ...
2
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2answers
91 views

Fibonacci Numbers and Legendre symbol

How to prove congruence below ? $$F_{p-\left( \frac{5}{p}\right)} \equiv 0 \pmod p$$ Where $\displaystyle \left( \frac{}{}\right)$ is legendre symbol, and $\displaystyle p$ is a prime number.
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3answers
63 views

Prove by induction that the Fibonacci sequence $≤ [(1+\sqrt{5})/2]^{n−1}$, for all $n ≥ 0$.

If $F(n)$ is the Fibonacci Sequence, defined in the following way: $$ F(0)=0 \\ F(1)=1 \\ F(n)=F(n-1)+F(n-2) $$ I need to prove the following by induction: $$F(n) \leq ...
0
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1answer
100 views

Fibonacci relation formula

There are three numbers a,b,c such that c=a+b. Let f(n) be n'th Fibonacci number,can we write f(a)+f(b) in terms of f(c) and c. If yes,how? I have tried deriving it using Binnets formula but did'nt ...
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1answer
140 views

Can the sum of different sets Fibonacci numbers be the same?

Is it possible to have two sets having at least one different element and the sum of Fibonacci of all elements be the same? As in, two subsets: ...
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0answers
914 views

Fibonacci sum of all subsets of an array

Given a multiset, suppose $S_1, S_2, S_3 ... S_{2^n-1}$ are the subsets of that multiset and the sum of the elements of $S_1$ is ${SS}_1$, similarly ${SS}_2$ and ${SS}_3$ etc. If I want to find out ...
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3answers
134 views

How many $a$-nary sequences of length $b$ never have $c$ consecutive occurrences of a digit?

Let $S(a,b,c): = \#\{a$-nary sequences of length $b$ without $c$ consecutive occurrences of a digit$\}$. For example, $S(2,n,3)$ would be the number of binary sequences of length $n$ without $3$ ...
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1answer
62 views

The 9 most significant digits in Fibonacci series (Project Euler 104)

With regards to project Euler, problem 104: https://projecteuler.net/problem=104 The essence of the question here is how to keep track of the 9 most significant digits of a Fibonacci series (Keeping ...
3
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1answer
1k views

Find the sum of Fibonacci Series

I have given a Set A i have to find the sum of Fibonacci Sum of All the Subset of A ...
2
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1answer
122 views

Calculating Irrationals raised to some Power modulo 1000000007 [closed]

Lets define a function F as $F(n) = 1+(\frac{1+{\sqrt 5}}{2})^n$ As per wolfram site, ${\sqrt 5}\%99991=10104$ As per wolfram site, ${\sqrt 5}\%1000000007=no\_solution$ I need to find the value of ...
2
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1answer
143 views

Relation between Fibonacci Numbers [closed]

Is there any relation between $f(a), f(b)$ and $f(a+b)$ where $f(n)$ is the $n$'th fibonacci number?
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0answers
13 views

Potential Function Runtime

The potential function of a Fibonacci Heap is Φ(H) = t(H) + 2m(H) CLRS states in Figure 21.2 ...
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1answer
59 views

Interesting question on Fibonacci numbers. [duplicate]

Ran across this interesting question about the Fibonacci numbers but quite unsure how to go about it, any ideas ?
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1answer
34 views

Weird informatic problem with Fibonacci numbers in which I have some troubles

I don't know what happended to this website but for months I am not able to connect me in it. As I understand it the website is closed. It is in this website I found this problem. Let $L$ be ...
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1answer
38 views

Proof a formula of the Fibonacci sequence with induction

It turns out that the Fibonacci sequence satisfies the following explicit formula: For all integers $F_{n} ≥ 0$, $F_{n} = \frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n+1} - ...
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0answers
310 views

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 ...
8
votes
2answers
841 views

Sum of inverse of Fibonacci numbers

If $F(n)$ is the nth Fibonacci number, How can I prove that: $$\sum_{i=1}^{\infty} \frac{1}{F(i)}\approx 3.36\, .$$