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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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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0answers
255 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
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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 ...
8
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4answers
814 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 $\sum_{k=1}^n f_k^3$ ?
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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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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 ...
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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 ...
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2answers
47 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 $n=1$ ...
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4answers
43 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 ...
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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}...
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3answers
100 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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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 ...
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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 ...
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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 ...
2
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1answer
72 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 calculating ...
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0answers
44 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
28 views

Fibonacci Sequence and Time taken

Consider the following (incomplete) java code, which calculates the Fibonacci numbers ...
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2answers
147 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 of ...
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5answers
1k 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
43 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
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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
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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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1answer
259 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
84 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
59 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 ...
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1answer
77 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
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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
129 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)\\ &fib(a+...
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2answers
307 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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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 ...
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1answer
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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
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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
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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 \bigg(\frac{1+\sqrt{5}}{2}\...
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1answer
101 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
147 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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3answers
135 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
77 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 ...
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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
133 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 $F(...
2
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1answer
148 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
62 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 ...
0
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1answer
51 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} - (\frac{1-\sqrt{5}}{2})^{n+1}...