# Tagged Questions

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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### 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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### 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,...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 $F_n$ ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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