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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Fibonacci Proof Using Induction

$$f(n) = \left\{\begin{matrix} 0 & n=1\\ 1 & n=2\\ f_{n-1} + f_{n-2} & n\geqslant 2\end{matrix}\right.$$ How can I prove by induction that $$f_{n} \leq \left ( \frac{1+\sqrt{5}}{2} ...
2
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2answers
53 views

Non-isomorphic combinatorial classes with growth rate equal to the golden ratio?

In a couple of weeks, I'm giving a talk about the growth rates of some combinatorial classes. In the introduction, I thought I'd present the class of tilings of a $n\!\times\!2$ strip with dominos, ...
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2answers
82 views

Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
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2answers
197 views

find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
4
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1answer
115 views

Proof/Disproof of property of perpendicular lines in the Fibonacci grid

In my struggle to try to make progress on this question, I am trying to find a proof or counterexample of the following, stronger, statement: Denote by $K$ the subset of the Gaussian integers such ...
2
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1answer
163 views

Cycles in the Generalized Fibonacci Sequence modulo a Prime

Suppose I have a fibonacci sequence 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 Now if I have a modulo 5 fibonacci sequence,it will look like ...
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1answer
60 views

Limit of a function not using Stirling's Approximation

I want to compute the following limit: $$\lim_{n\to\infty} \frac{\left(\frac{e}{F_{n+1}}\right)^{F_{n+1}} F_{n+1}!}{\left(\frac{e}{F_n}\right)^{F_n} F_n!},$$ where $F_n$ is the $n$th Fibonacci ...
1
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1answer
62 views

What is the broader name for fibonacci and lucas sequences?

Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be: $$x_0 = 53$$ $$x_1 = 62$$ $$x_n ...
1
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1answer
114 views

Register Machine on Fibonacci Numbers

This problem is easy to understand but I am struggling to come up with any solutions. According to Wikipedia a register machine is a generic class of abstract machines used in a manner similar to a ...
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1answer
27 views

Generalisation of Fibonacci

Somehow a generalisation of the fibonacci numbers, do numbers created by the formula $ F(n) = F(n-1) + [F(n-1)-F(n-2)+F(n-3)-F(n-4)+F(n-5)-F(n-6).....]$ with $F(1) = 1$ have a specific name?
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1answer
47 views

Fibonacci proof by Strong Induction

Prove by strong induction that for a ∈ A we have $F_a + 2F_{a+1} = F_{a+4} − F_{a+2}.$ $F_a$ is the $a$'th element in the Fibonacci sequence
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1answer
86 views

Simplifying Sigma Notation

I am working on the proof on showing the ratio of two consecutive Fibonacci numbers converges to the golden ratio to explain to a student I am tutoring. I am getting to some confusion in a ...
-1
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1answer
64 views

Application of convergence of Fibonacci series

'There are infinite prime numbers' is a fact that can be deduced by 'reciprocal of primes diverges' statement, so from this can we deduce the fact that --> 'there are finite Fibonacci numbers in ...
6
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0answers
425 views

A conjecture about Lucas series

Let $L_n$ be the Lucas series: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}(n>1).$ $p$ is a prime number and $p\equiv3,7\pmod {20}$, hence $\exists x,y\in \mathbb Z:2p=x^2+5y^2.$ Is it true that ...
6
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0answers
270 views

Construction of generating function from identity

I am trying to solve identity involving binomials and fibbonaci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose ...
4
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0answers
62 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows that ...
3
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0answers
120 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? ...
2
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0answers
37 views

Summation involving Fibonacci numbers

Find: $$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{F_{2k}F_{n-k}}{10^n} $$ where $F_n$ is $n$-th Fibonacci number.
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0answers
80 views

Why is the reciprocal of the second Fibonacci number negative?

The second Fibonacci number is 1, so it's reciprocal should be 1, right? Why is it that I get $-1$ when I plug in $2$ for n in the reciprocal of Binet's equation ...
2
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0answers
282 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
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0answers
34 views

Could Fibonacci numbers be related to Riemann zeros?

this is my question can tghe fibonacci numbers $$ F_{n+2} =F_{n+1} +F_{n} $$ be related to the zeros of the Riemann zeta function ?? i heard that in the webpage ...
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0answers
30 views

Fibonacci series mod a number

I'm trying to write a program with an input of numbers $n$ and $k$ (where $n<10^{20}$ and $k<10^9$), where I compute fib[n] % k. What is a good FAST way of computing this? I have read many ...
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0answers
32 views

Question on Proof that the Fibonacci Word is Sturmian

I am currently reading a text where it is proved that the infinite Fibonacci Word $u$ defined as the limit of the sequence $$ u_n = \varphi^n(0) $$ where the morphism is given by $\varphi(0) = 01, ...
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0answers
50 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
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0answers
86 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
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0answers
37 views

Lucas Number Equivalent of the Pisano Period?

I'm doing some work with the Pisano Period, and it's leading to also talk about the Lucas numbers $mod \ m$, and their period. Is there a specific notation already designated? Sticking with using a ...
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0answers
199 views

Calculating pisano periods for any integer

I recently stumbled across this SPOJ question: http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the pisano period of a number. After I researched my way through the web, I found ...
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0answers
64 views

Simplify Fibonacci Power Series

I am working on an algorithm to count the number of models for Exactly One in Three SAT (X3SAT) instances. It is known that a chain of X3SAT clauses of length $c$ has $F(c+3)$ satisfying assignments ...
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0answers
42 views

Strange equality of the operator E($Eu_n=u_{n+1}$)

The operator $E$ is defined as $Eu_n=u_{n+1}$. I encountered a strange equality. when I tried out Let $u_n$ represent a series such that $$u_{n+2}=u_{n+1}+u_n. \tag{$\star$}$$ Or ...
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0answers
102 views

Is this “Elegant” algorithm for logarithm by Zeckendorf representation, the same as an 'efficient' algorithm?

The algorithm here which computes the exponent $b$ given a base $a$, and given $n$ = $a$^$b$, appears no better to me than simply counting the number of times we divide $n$ through by the base $a$ ...
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0answers
101 views

The zeta-function of Fibonacci sequence?

I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some ...
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0answers
187 views

Fibonacci and Lucas relations

We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following ...
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0answers
47 views

how would you represent the tribonacci spiral?

As we all know, the Fibonacci Sequence has a very famous spiral representation. Here we can see it. Now it's easy to see the fibonacci is the sum of the 2 squares that compose it to form a ...
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0answers
225 views

fibonacci numbers mod some prime number

Moderator Note: This is a current Code Chef challenge question. When the current challenge ends on 15 October 2013 this question will be unlocked. I have prime numbers ($\geq11$) and of the form ...
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0answers
70 views

Fibonacci sequense, problem od division

How to show that $7\mid F_m\Longrightarrow 8\mid m$ and $4\mid F_m\Longrightarrow 6\mid m$, knowing that (I) Two consecutive terms in the Fibonacci sequence are relatively prime. (II) In ...
0
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0answers
82 views

Finding the kth n-anacci number

At this wolfram link the formula for the kth n-anacci number is given: http://mathworld.wolfram.com/Fibonaccin-StepNumber.html#eqn8 (Eq. 4) Not sure if I understand correctly. If I want the fifth ...