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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259 views

Solving Fibonaccis Term Using Golden Ratio Convergance

While solving this problem, I discovered that there is a relationship between the Fibonacci sequence and the golden ratio. After I got the correct answer via brute force, I discovered this ...
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1answer
1k views

Smallest Fibonacci number having a common factor with a given number

We have a number $k$ and we have to find the smallest Fibonacci number that has common factor with it(except $1$). We also have $2 \leq k \leq 1,000,000$. The required Fibonacci number is guaranteed ...
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2answers
176 views

Count the number of paths in the Graph $P_3$. Provide a Proof by Induction using the Fibonacci sequence.

Consider the graph $P_3$ : $n_1$$ \rightarrow$ $n_2$$\rightarrow$ $n_3$$\rightarrow$ $n_4$ we count 6 paths of length k=1, namely: $n_1$ $\rightarrow$ $n_2$ $n_2$ $\rightarrow$ $n_3$ ...
3
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2answers
3k views

sum of even-valued and odd-valued Fibonacci numbers

I was solving the Project Euler problem 2 *By starting with 1 and 2, the first 10 terms of Fibonacci Series will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... Find the sum of all the even-valued terms ...
2
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1answer
137 views

Finding n in Fibonacci closed loop form

The nth term of the Fibonacci series is given by $F_{n}$=$\Big\lfloor\frac{\phi^{n}}{\sqrt{5}}+\frac{1}{2}\Big\rfloor$ How do you get the following expression for n from this? ...
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5answers
4k views

Why does the Fibonacci Series start with 0, 1?

The Fibonacci Series is based on the principle that the succeeding number is the sum of the previous two numbers. Then how is it logical to start with a 0? Shouldn't it start with 1 directly?
3
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1answer
173 views

Fibonacci numbers moduli

I have made some observation on very interesting material on Fibonacci series. I need some help in proving them mathematically. We can observe that the periodicity of Fibonacci numbers modulo m, ...
20
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1answer
330 views

The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized

The evaluation, $$\sum_{n=0}^\infty \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}=\left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2$$ was recently asked in a post by Chris here. I ...
10
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3answers
244 views

Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$

Evaluate the sum: $$\sum_{n=0}^{\infty} \frac{1}{F_{(2^n)}}$$ where $F_{m}$ is the $m$-th term of the Fibonacci sequence. I need some support here. Thanks.
7
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1answer
744 views

How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$

I am trying to prove it by induction, but I'm stuck $$\mathrm{fib}(0) = 0 < 0! = 1;$$ $$\mathrm{fib}(1) = 1 = 1! = 1;$$ Base case n = 2, $$\mathrm{fib}(2) = 1 < 2! = 2;$$ Inductive case ...
4
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1answer
189 views

Golden parallelepiped

Define a golden parallelepiped as a $d$-dimensional box with side lengths $(1, \phi, \phi^2, \ldots, \phi^{d-1})$, where $\phi$ is the golden ratio: ...
6
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1answer
271 views

Finding ($2012$th term of the sequence) $\pmod {2012}$

Let $a_n$ be a sequence given by formula: $a_1=1\\a_2=2012\\a_{n+2}=a_{n+1}+a_{n}$ find the value: $a_{2012}\pmod{2012}$ So, in fact, we have to find the value of ...
3
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0answers
153 views

Which starting conditions for the Fibonacci sequence, gives most primes

I found the following question (at http://aperiodical.com/2012/05/matt-parkers-twitter-puzzle-25-may/): If you start the Fibonacci sequence 2,1 instead of 1,1 do you get more or fewer primes? ...
4
votes
1answer
309 views

What is the next “Tribonacci-like” pseudoprime?

Given the three roots of $x^3=x^2+x+1$. Then we get the tribonacci-like sequence, $B_n = x_1^n+x_2^n+x_3^n = 3, 1, 3, 7, 11, 21, 39, 71, 131,\dots$ where $B_n = B_{n-1}+B_{n-2}+B_{n-3}$, and the ...
12
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1answer
310 views

Why do the Fibonacci numbers recycle these formulas?

The Fibonacci numbers $F_n = 0, 1, 1, 2, 3, 5, 8, 13, \dots$ obey the following recurrence relations, $ \begin{aligned} &F_{n}-\;F_{n-1}-F_{n-2} = 0\\[1.5mm] &F_{n-1}^3-F_{n}^3-F_{n+1}^3 = ...
4
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2answers
520 views

Use of the Reciprocal Fibonacci constant?

The Reciprocal Fibonacci constant ($\psi$) is defined as $$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$ where $F_{k}$ is the $k^{th}$ Fibonacci number. The irrationality of $\psi$ has been proven. ...
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2answers
401 views

A game: Fibonacci sequences and probability.

Let's play a game. You have two biased coins: coin A has a $0.4$ and $0.6$ for H and T probability and coin B has the opposite ($0.6$ and $0.4$ for H and T). These coins must be flipped one at a time, ...
2
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2answers
626 views

Why is fibonacci coding useful?

I have read this wiki article but it seems not very clear to me. Why should we ever use fibonacci coding in data compression if even regular binary coding always gives better results? I mean, it seems ...
6
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2answers
187 views

Prove that $\sum\limits_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= \sum\limits_{n=0}^{\infty}\frac{1}{2^{n}}$

I came up with this identity in high school, and I can't remember how I proved it :P Does anyone know how I would go about doing this? $$\sum_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= ...
3
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1answer
198 views

If $m$ and $n$ are positive integers, then $(F_m,F_n)=F_{(m,n)}$.

Edit: The $F$'s are Fibonacci numbers. I need an idea on how to show the following: If $m$ and $n$ are positive integers, then $(F_m,F_n)=F_{(m,n)}$. I believe that using the fact that ...
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5answers
527 views

Showing that $f_{2n+1}=f_{n+1}^2+f_n^2$.

I am trying to solve the following exercise: Let $f_1=1$, $f_2=1$, $f_{n+1}=f_n+f_{n-1}$, where $n\in\mathbb{N}$. Show that $f_{2n+1}=f_{n+1}^2+f_n^2$. I have not had much progress, but this is ...
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2answers
543 views

Using linear algebra, how is the Binet formula (for finding the nth Fibonacci number) derived?

If possible, please refrain from any type of proof besides linear algebra. So, using the recursion formula $F_{n+1} = F_{n-1} + F_n$, for $n\gt 1$, and where $F_0 = 0$ and $F_1 = 1$, and the Fibonacci ...
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0answers
82 views

Lucas numbers theory confused [duplicate]

Possible Duplicate: Prove this formula for the Fibonacci Sequence How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$ How do I go about ...
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3answers
305 views

Another way to go about proving Binet's Formula

As I showed in another question of mine, it is easy to prove that $$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$ given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$. Now, extending $(1)$ ...
7
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3answers
541 views

How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$

How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$ where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio?
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1answer
423 views

(Long) Fibonacci Sequence Question

I am working on a rather long question. I am going to write out the question, and what I've come up with so far. I'm not sure if the question has too many parts to receive answers but I will ...
5
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4answers
16k views

What do Subscripted numbers in an equation mean?

$F_n= F_{n-1}+ F_{n - 2}$ I know that when a number is superscripted it means "to the power of", but what does the subscript mean?
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1answer
127 views

“Non-commutative” Recurrence relation $ C_m = \alpha_m C_{m-1} + \beta_m C_{m-2} $

I have a problem, which is probably quite trivial. Consider a recurrence relation of the form $$ C_m = \alpha_m C_{m-1} + \beta_m C_{m-2}, $$ where the coefficients $\alpha_m$ and $\beta_m$ are ...
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3answers
593 views

Infinite Series: Fibonacci/ $2^n$

I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner) In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... each term ...
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4answers
2k views

Fibonacci's final digits cycle every 60 numbers

How would you go about to prove that the final digits of the Fibonacci numbers recur after a cycle of 60? References: The sequence of final digits in Fibonacci numbers repeats in cycles of 60. The ...
6
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3answers
199 views

Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$

Prove: $\binom{n}{0}F_0+\binom{n}{1}F_1+\binom{n}{2}F_2+\cdots+\binom{n}{n}F_n=F_{2n}$; I was stuck with this question for a while... Help me please!!! Thanks!!!
3
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2answers
129 views

Binet's Formula. An operational approach.

I read quite a while ago this proof of Binet's formula. ( I am not 100% sure this is the way it was presented, but it gives an idea. I'm not approving of this method or saying it is correct.) Let ...
7
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2answers
285 views

Fibonacci numbers with largest prime factor appearing more than once

$F_6=2^3$ and $F_{12}=2^43^2$. Is there an $n>12$ such that $F_n=p^2k$ with $p$ prime and $k$ is $p$-smooth?
7
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1answer
656 views

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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3answers
176 views

How can I show that $\sum\limits_{n=1}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}$ is algebraic?

Show that $$\sum_{n=1}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}$$ is algebraic. More specifically, solve this and get exact values. Then use the result to evaluate $$\sum_{n=0}^\infty ...
30
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3answers
648 views

Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
7
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3answers
877 views

Prove by induction Fibonacci equality

[question:] Prove by induction that the i th Fibonacci number satisfies the equality $$F_i=\frac{\phi^i-\hat{\phi^i}}{\sqrt5}$$where $\phi$ is the golden ratio and $\hat{\phi}$ is its conjugate. ...
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2answers
302 views

Generalized Fibonacci Sequence Question

The Fibonacci Sequence is defined as the recurrence $a(n)=a_{n-1}+a_{n-2}$ where $a(0)=0$ and $a(1)=1$. Today, I was bored so I considered the sequence $a(n)=\sqrt{a_{n-1}}+\sqrt{a_{n-2}}$. Ten ...
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8answers
897 views

Need help deriving recurrence relation for even-valued Fibonacci numbers.

That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$ Empirically one can check that: $a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$. If $f(n)$ is ...
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1answer
499 views

Why Fibonacci numbers are too often found on nature?

Fibonacci number has something to do with natural growth. Though the function is very straight forward, we see this in nature. Does nature follow the function or its the simplified model of the ...
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4answers
270 views

Generalized Fibonacci sequences

Why Fibonacci sequence start at $0$, Tribonacci sequence with $0,0$, Tetranacci with $0,0,0$, etc. [ref OEIS] Has any good reasons for that? These sequences arise in generalization of Pascal Triangle ...
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1answer
103 views

Recovering two first terms from sequence $f(n)=f(n-1)+f(n-2)$

Having very simple sequence $f(n)=f(n-1)+f(n-2)$ and having $n-th$ term given how can we calculate from which first two terms this $n-th$ term came? I know the answer can be not unique so highest ...
2
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2answers
650 views

Recurrence for a lagged Fibonacci sequence

I know how to do the matrix for the standard $f(n) = f(n-1) + f(n-2)$ relationship, but what if it's piecewise? For instance, $f(1)$ through $f(30)$ have some preset values, and then for $f(31)$ ...
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2answers
3k views

How to find the closed form to the fibonacci numbers? [duplicate]

Possible Duplicate: Prove this formula for the Fibonacci Sequence How to find the closed form to the fibonacci numbers? I have seen is possible calculate the fibonacci numbers without ...
3
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3answers
167 views

Prove for Fibonacci numbers: $3\mid f(n) \iff 4\mid n$

Let $f(n)$ be the $n$th Fibonacci number. Prove that $$3\mid f(n) \iff 4\mid n$$ I tried to use induction to prove it but I couldn't continue when I reached $n+1$ case.
13
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3answers
712 views

Prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$

We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact, $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
8
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4answers
2k views

Proof by Induction: Alternating Sum of Fibonacci Numbers [duplicate]

Possible Duplicate: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer This is a homework question so I'm looking to just be nudged in ...
1
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3answers
400 views

Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer

Letting $f_n$ be the Fibonacci numbers, show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer. Just some homework help. Need to prove. Thank you in ...
5
votes
4answers
2k views

How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
5
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2answers
236 views

Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) ...