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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Hadamard's product of Fibonacci generating functions.

$F(s) = \frac{1}{1-s-s^2}=\sum_{n\geq0}F_ns^n$. I want to calculate $F(s) \circ F(s) = \sum_{n\geq0}F_{n}^2s^n$. I have tried using Binet"s formula, but problem remains unsolved.
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1answer
43 views

Fibonacci Number Formula for nth term [duplicate]

Hey is there any known combinatorial formula for nth fibonacci number? (n+1)th fibonacci number is given by summation of r=0 to (round)n/2:C(n-r,r) Can someone verify the formula?Help!
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1answer
41 views

Help with how to prepare the inductive step of a strong induction exercise.

I have the following exercise: "Use strong induction to prove that $f_1^2 + f_2^2 + \cdots + f_n^2 = (f_n)(f_{n+1})$ where $f_n$ in the nth Fibonacci number." This is what I have done: Fibonacci ...
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0answers
28 views

Can't find ANY golden ratio in the schroder house…

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to ...
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90 views

Proving $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $ for Fibonacci numbers

How can I prove the following identity about the Fibonacci numbers by using matrices or determinants? $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $
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28 views

Prove that for any power function $f_n = c^{n}$, the number of decimal digits of $ f_{10^n}$ is given by $10^{n}log_{10}c$

I am reading this page about some interesting properties of the Fibonacci numbers: http://mathworld.wolfram.com/FibonacciNumber.html The following is said: The numbers of Fibonacci numbers less ...
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2answers
65 views

Given that fib(n)=fib(n-1)+fib(n-2) for n>1 and given that fib(0)=a, fib(1)=b (some a, b >0)

Given that fib$(n)$=fib$(n-1)$+fib$(n-2)$ for $n>1$ and given that fib$(0)=a,$ fib$(1)=b$ $($some $a, b >0)$ which of the following is true? fib$(n)$ is : Select one or more: a. $O(n)$ b. ...
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0answers
55 views

Identify fibonacci sequences from a set of data

Let there be a set of increasing order integer data ${a_1, a_2, a_3, a_4, ...}$. given the increasing infinite sequence of integers, how can we determine whether there is an infinite subsequence which ...
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40 views

Fibonacci polynomial summation

The $n^{th}$ value of a polynomial ($S_n$) of order $k$ is a polynomial in $x$ is given by : $\left(\frac{S_n}{x^n}\right)$= $\sum_{j=0}^n \left(\frac{{F_j}^k}{x^j}\right)$ Where ${F_j}$ is the ...
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1answer
21 views

Show that $\forall n \in \mathbb{N}: n$ can be written as $F_{n+1}$ different sums of ones and twos

Show that $\forall n \in \mathbb{N}: n$ can be written as $F_{n+1}$ different sums of ones and twos where the order matter. Presumably, mathematical induction can be leveraged here. Step 1: Show ...
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2answers
48 views

Applying the mean value theorem to the closed form of the Fibonacci sequence?

Is it possible to apply the mean value theorem to the closed form of the Fibonacci sequence for the 7 numbers starting at 1 and ending with 13 (inclusive)? It's been a LONG time since I studied ...
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1answer
52 views

Deduce a series formula for product of Fibonacci numbers.

Start with the arbitrary pair of Fibonacci numbers $F_{n+1}$, $F_n$ and apply the Euclidean Algorithm to it. Deduce a series formula for the product $F_{n+1}F_n$. I use the formula, $F_{n+1} = F_n + ...
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1answer
59 views

Proving a function by induction [duplicate]

Let $f(n)$ be the function defined by $$ f(n) = \frac 1{\sqrt{5}} \left[ \left(1+\sqrt{5}\over2\right)^n- \left(1-\sqrt{5}\over2\right)^n \right] $$ How do you prove that $f(n) = f(n+2) - f(n+1)$ ...
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4answers
102 views

Relationship between Fibonacci's secuence and $x^2 - x - 1$.

On the end of Apostol's Mathematical Analysis' first chapter, one can find the following exercise (and I paraphrase): Prove that the $n$-th term of the Fibonacci sequence is given by $$x_n = ...
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2answers
187 views

Fibonacci even numbers formula

i found a general formula in any given set of Fibonacci numbers ,to find the next given even number we can use the formula E*4 + Eo where E is the given even number Eo is the even number ...
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3answers
93 views

Lucas numbers and fibonacci

This is a question straight from the Applied Combinatorics book. Suppose that chairs are arranged in a circle. Let $L_n$ count the number of subsets of $n$ chairs which don't contain consecutive ...
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99 views

Are there infinite Fibonacci primes if and only if there are infinite Fibonacci numbers that are Fibonacci pseudoprimes?

One of the open questions about the Fibonacci numbers is if there are infinite prime numbers inside the Fibonacci sequence. I wonder if a good approach would be trying to know first if there are ...
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2answers
42 views

Prove the sum of the even Fibonacci numbers

Let $f_n$ denote the $nth$ Fibonacci number. Prove that $f_2\:+\:f_4\:+...+f_{2n}=f_{2n+1}-1$ I am having trouble proving this. I thought to use induction as well as Binet's formula where, ...
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189 views

LCM of Fibonacci numbers

$\newcommand{\lcm}{\operatorname{lcm}}$There is a nice property of Fibonacci numbers which says that: $$\gcd(F_{a_1}, \ldots, F_{a_n}) = F_{\gcd(a_1, \ldots, a_n)}$$ I am curious is there anything ...
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1answer
54 views

Showing that $f_n$ is the number closest to $\frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n$ [duplicate]

I want to prove that the the $n$th Fibonacci number $f_n$ is the integer closest to $\frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n$. What would be a rigorous way to go about this? I assume I'll have to ...
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26 views

Hankel determinant involving Fibonacci numbers

Let $F_n$ denote the nth Fibonacci number, with $F_1 = F_2 = 1$. Denote by M(n) the nxn Hankel matrix with $i,j $ entry $F_{i+j-1}^{n-1}$, where i and j range from 1 through n. Finally, let d(n) = ...
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1answer
108 views

Connection between Euler's totient function and Fibonacci numbers

For a sequence $(a_n)$ of natural numbers define $\alpha(n):=\min\{m\in\mathbb{N}:n|a_m\}$ whenever it exists. Thus $\alpha(n)$ is the first index $m$ such that $n$ divides $a_m$. Now define the ...
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1answer
47 views

Convergence of fibonacci quotient $\frac{f_n}{f_{n+1}}$

I know that $\frac{f_{n+1}}{f_{n}}$ converges against $\phi = \frac{1+\sqrt{5}}{2}$. The question i want to to ask you is if the following conclusion is correct, I mean i know that if we have two ...
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1answer
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What is the probability that a given a positive number, will be found in the space of shifted fibonacci sequences?

We are given a space of shifted Fibonacci sequences, Fk, Fk+1, Fk+2, Fk+3, Fk+6, Fk+8, Fk+9, Fk+10, Fk+10, Fk+11….. Given a number,n, what is the probability of this number within this space? And ...
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187 views

when $F_n^2+F_m^2$ is a square for fibonacci numbers

This is a curiosity question I'm trying to solve a Diophantine equation and I need some results about fibonnacci numbers, I encountered this problem: For which couple of integers $(n,m)$ the ...
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3answers
219 views

How many times a positive number can be found in shifted Fibonacci Sequences?

Given a positive number, how many times can this number be found in shifted fibonacci sequences? ...For example...Number 11, can be present seven times in total, in Fn+3, Fn+6, Fn+8, Fn+9, Fn+10, ...
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2answers
77 views

How to find a Fibonacci number that is divisible by $x$?

I'm looking for an algorithm that is better than just checking every number in the Fib Sequence for divisibility. Example: Find the first Fib number that is divisible by $x=223321$, with no ...
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2answers
114 views

Find a formula for the nth Fibonacci Number [duplicate]

So I'm being asked to find a formula for the nth fibonacci number. I know the answer is $$x_{n}=\frac{(1+5^{1/2})^{n} -(1-5^{1/2})^n}{\sqrt{5}2^n}$$ However I don't really know how to get there. ...
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2answers
56 views

Seeking combinatorial proof for $F_{n+1} -1=\sum\limits_{k=0}^{n-1} F_k$

In order to give a combinatorial proof for this equation, we need to find what these two count for. But I don't know what they count for and how I can pivot the RHS to show that it actually counts ...
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94 views

Prove function is Fibonacci sequence

Stuck on how to finish a question I'm working on. Have to find the number of bitstrings of length n with no odd length maximal runs of ones. For example, when n=3 there are three such bitsrings: 011, ...
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1answer
52 views

Lucas Number Sequence

Can anyone help me in this question: Define $ (b_n)$ as $b_1= 1,b_n=a_{n+1} - a_n $ for $ n\ge 2$, where $ a_n $ is the Fibonnaci series. This sequence is known as the sequence of Lucas numbers. ...
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178 views

Show that there are infinitely many integers such that $ \binom{m}{n-1} = \binom{m-1}{n} $

This question comes from the 1st Brazilian's IMO TST of 2004. I have found no solutions of it online, though I have developed one. After getting to $ mn = (m-n)(m-n+1) $, my solution relies on the ...
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1answer
37 views

Fibonacci Numbers, show $F_n$ $>=$ $2^{n/2}$ for n $>=$ 6.

I want to show that for the fibonacci numbers, $F_n$ $>=$ $2^{n/2}$ for n $>=$ 6. My thought was to prove this via induction. I showed the base case is true for $F_n$, n=6 and 7. I assumed ...
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2answers
321 views

If $ab-1$, $bc-1$, $ca-1$, $ab-a-b+c$, $bc-b-c+a$, $ca-c-a+b$ are perfect squares, then are $ab+a+b-c$, $bc+b+c-a$, $ca+c+a-b$ also perfect squares?

About a month ago, a friend of mine taught me that there exist many sets of three positive integers $(a,b,c)$ where $a\not=b,b\not=c$ and $c\not=a$ such that each of $$ab-1,\ bc-1,\ ca-1,\ ab-a-b+c,\ ...
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1answer
53 views

Proving the Fibonacci identity $\sum_{i=1}^n f_i^2=f_nf_{n+1}$ by induction [duplicate]

I am having troubles with a proof question. Prove that for any $n\ge1$, $\sum_{i=1}^n f_i^2=f_nf_{n+1}$, where $f_n$ is the $n$'th Fibonacci number. I have the base case and the induction ...
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Of fibonomials, pellonomials, and tribonomials, etc

I. Linear recurrence with order 2 Given the Fibonacci numbers $F_n$, we have $$\begin{aligned} &F_n+F_{n+1}-F_{n+2}=0\\[1mm] &F_n^2-2F_{n+1}^2-2F_{n+2}^2+F_{n+3}^2=0\\[1mm] ...
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Generalizing the Fibonacci identity $F_{2n}=-F_{n-1}^2+F_{n+1}^2$

Using an integer relations algorithm, we get, $$F_{2n}=-F_{n-1}^2+F_{n+1}^2$$ $$6F_{4n}= -F_{n-2}^4-3F_{n-1}^4+3F_{n+1}^4+F_{n+2}^4$$ The pattern of the subscripts is clear. Expressing the ...
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2answers
101 views

formula for logarithmic spiral on a linear level

I am trying to plot the contents of a circle, which include geometric elements and spirals, on a linear graph. For example, take a circle, take the beginning and the end and make it straight. What ...
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2answers
54 views

Fibonacci recursive algorithm yields interesting result

After writing a program in Java to generate Fibonacci numbers using a recursive algorithm, I noticed the time increase in each iteration is approximately $\Phi$ times greater than the previous. ...
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2answers
138 views

Need help understanding Fibonacci Fast Doubling Proof

From this website, http://www.nayuki.io/page/fast-fibonacci-algorithms (fast doubling proof close to the bottom of the page). I have understood the proof for the most part but I am struggling to see ...
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Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
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203 views

A number $N$ is a $k$-nacci number if and only if …

For $k\ge 2\in\mathbb N$, one can define the $n$-th $k$-nacci number $f_k(n)\ (n=0,1,\cdots)$ as $$f_k(0)=f_k(1)=\cdots=f_{k}(k-2)=0,\ \ ...
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1answer
34 views

How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about ...
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1answer
75 views

Are there 3D geometric proofs of Fibonacci identities?

There is a significant number of identities involving Fibonacci numbers that can be proven in a sort of geometric way, as it is shown in the following picture: However, I couldn't find any such ...
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71 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
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1answer
60 views

Closed form as sum and combinatorial of Fibonacci numbers

How can I prove that the Fibonacci numbers that are defined as $F_n=F_{n-1}+F_{n-2}, \; n \geq 2$ and $F_0=0,\ F_1=1,\ F_2=1$ have the form: $$F_n=\sum_{k=0}^{n-1} \binom{n-1-k}{k}, \; n\ge 2 $$ I ...
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0answers
151 views

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$?

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$? By better comparison series than $\sum_0^\infty2^{-k}$ we mean a series $\sum c_k$ s.t. ...
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1answer
73 views

What is wrong with the following argument involving Fibonacci and Lucas numbers?

The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$. What is wrong with the following argument? Assuming $L_n = F_n$ for $n = ...
3
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4answers
231 views

A calculus proof for the general term of the Fibonacci sequence [duplicate]

Let $a_0=1$,$a_1=1$ and $a_n=a_{n-1} + a_{n-2}$ for $n \geq 2$, I would like to prove: $$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n + 1}- \left(\frac{1-\sqrt{5}}{2}\right)^{n + ...
3
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4answers
48 views

Prove $F^2_{n+1} - F_nF_{n+2} = (-1)^n$

This is a question about Fibonacci sequences, a sequence in which the previous terms build up upon the current term (e.g. $F_1 + F_2 = F_3$ where $F_1 = F_2 = 1$). How would I go about proving ...