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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2answers
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Relationship between golden ratio powers and Fibonacci series

Can anyone prove the following equation? ($F_n$ is the $n$th element of Fibonacci series and $n \in N$.) $\phi = 1 \times \phi + 0$ $\phi^2 = 1 \times \phi + 1 $ $\phi^3 = 2 \times \phi + 1 $ ...
4
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0answers
112 views

Connections between Fibonacci and natural numbers

Here are some known facts about the Fibonacci numbers and then some questions regarding them . 1.Carmichael's theorem : For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of ...
8
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1answer
147 views

How to evaluate this infinite product ? (Fibonacci number)

Let $F_n$ be Fibonacci numbers. How to evaluate $$\prod_{n=2}^\infty \left(1-\frac{2}{F_{n+1}^2-F_{n-1}^2+1}\right)\text{ ?}$$ It seem like that $$\prod_{n=2}^\infty ...
15
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5answers
846 views

Fibonacci number identity.

How do I see that $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$, $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?
24
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1answer
568 views

Parabolas in sequences of digits from the Fibonacci sequence

In preperation for an exam, I was studying Haskell. Therefore I was solving an old assignment where you had to define the fibonacci series. After solving the task (see 1] for source code) and ...
-1
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1answer
35 views

Adding two variables with subscripts [closed]

What is the explanation to why $x_{3k} + x_{3k+1}$, is equal to $x_{3k+2}$. Isn't that incorrect because there is no value 1 in the subscript $x_{3k}$? I saw this in a prove in ...
0
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0answers
31 views

Prove that for every $k$ there exist fibonnaci number that ends with $k$ zeros.

Let $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Prove that for every $k$ there exist $F_m$ that ends with $k$ zeros. I tried using pigeonhole principle, but with no effect.
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3answers
1k 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 $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each ...
3
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1answer
103 views

Can the Fibonacci sequence be written as an explicit rule?

When I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = ...
2
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1answer
54 views

Is there a proven formula for Pisano periods (Fibonacci numbers taken modulo $n$)?

I've seen on wikipedia this formula: $$\pi_k(p^n) = p^{n-1}\pi_k(p) $$ It says that the formula holds unless p is k-Wall-Sun-Sun prime, or k-Fibonacci-Wieferich prime, that is, $p^2$ divides ...
11
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1answer
224 views

Fibonacci numbers and the nontrivial zeros of the Riemann zeta function

Is this a mathematical coincidence? For $n=1,\dots,7$: $$ \left\lfloor \prod_{k=1}^n \arg\left(\rho_k\right)\right\rfloor = F_{n+1}, $$ where $\arg$ is the complex argument, $\rho_n$ is the $n$th ...
4
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2answers
2k views

Finding index of a Fibonacci number: any mathematical solution possible?

The problem: Given a Fibonacci number,find its index. I am aware of the standard solution 'generate-hash-find'. I am just curious if there is ...
3
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4answers
95 views

Fibonacci proof question $\displaystyle \sum_{i=1}^nF_i = F_{n+2} - 1$

The sequence of numbers $F_n$ for $n \in N$ defined below are called the Fibonnaci numbers. $F_1 = F_2 = 1$, and for $n \geq 2$, $F_{n+1} = F_n + F_{n-1}$. Prove the following facts about the ...
17
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3answers
990 views

How to 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} = ...
2
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1answer
50 views

Let $a^n = a^{n - 1} + a^{n -2}$. Show that for any $A, B$, $F(n) = Aa^n + Bb^n$ satisfies Fibonacci recurrence relation.

$$\begin{align*} F(n) &= Aa^n + Bb^n\\ &= A(a^{n-1}+a^{n-2}) + B(b^{n-1}+b^{n-2}) \\ &= Aa^{n -1} + Aa^{n-2} + Bb^{n -1} + Bb^{n-2}\\ &= a^{n -1} (A + A^{a-1}) + b^{n - 2} (B + bB) ...
0
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1answer
37 views

Does the smallest prime factor of a Fibonacci number appear in the Fibonacci sequence?

I thought of a way to tackle the problem of knowing whether there are infinitely many Fibonacci primes or not and this question came to my mind: does the smallest prime factor of any Fibonacci number ...
2
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1answer
72 views

strange fibonacci recurrence

As it is well known fibonacci numbers satisfy the recurrence relation $$F_{n}=F_{n-1}+F_{n-2}$$ with initial conditions $F_{0}=0$ and $F_{1}=1$. While playing around with numbers,I noticed the ...
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3answers
73 views

How to prove a specific sequence is Fibonacci's with no prior knowledge nor trial and error?

Let $n$ be a positive integer and let $s_n$ be the number of increasing sequences of integers, alternatingly even and odd, starting with $0$ and ending with $n$. E.g. for $n=3$ we only have the two ...
7
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1answer
85 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
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6answers
134 views

Expression for negating every other odd number index

Is there a way to have an iterative expression that negates every other odd number index (starting from 3)? Basically, I am trying to write a generative expression that will give me value, given ...
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3answers
1k views

People sitting in a circle chewing gum

Ten people are sitting in a circle of ten chairs, chewing gum. Each person spits out his or her gum and places it either under his or her own chair or under an immediately adjacent chair. How many ...
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4answers
12k views

The generating function for the Fibonacci numbers

Prove that $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$ The coefficients are Fibonacci numbers, i.e., the sequence $\left\{1,1,2,3,5,8,13,21,...\right\}$.
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2answers
74 views

Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$

I have the following problem: Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$ Where $F_n$ is the $nth$ Fibonacci number. Proof Basis $n = 6$. $F_6 = 8 \geq 2^{0.5 \cdot ...
3
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2answers
83 views

Indexes of prime Fibonacci numbers

I found this on Mathworld, but I can't seem to find any proof, either on StackExchange, nor any other site: Why do all Fibonacci primes, except for $F_4=3$, have prime indexes (with $F_0=0$)? My ...
2
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2answers
74 views

Fibonacci Cyclic Pattern [duplicate]

I want to show the Fibonacci numbers are cyclic in mod n. I have tried some small values for n and I can see this is the same. In terms of a proof, I'm thinking of using the pigeonhole principle of ...
2
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1answer
58 views

Proving the Fibonacci sum $\sum_{n=1}^{\infty}\left(\frac{F_{n+2}}{F_{n+1}}-\frac{F_{n+3}}{F_{n+2}}\right) = \frac{1}{\phi^2}$ and its friends

In this article, (eq.92) has, $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+1}F_{n+2}} = \frac{1}{\phi^2}\tag1$$ and I wondered if this could be generalized to the tribonacci numbers. It seems it can ...
4
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2answers
194 views

Some infinite series with Fibonacci numbers

An interesting problem is to prove that: $$ \sum_{n=1}^\infty \frac{F_{2n}}{n^2 \binom{2n}{n}}=\frac{4\pi^2}{25 \sqrt 5}. $$ I know the proof, which uses the fact that ...
4
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0answers
84 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
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1answer
119 views

Is there a pattern of the length between one even Fibonacci number and another?

I had seen a math problem asking for the sum of all even Fibonacci numbers up to 4 million, but I still need to know this: Is there an obvious pattern of the distance between a even Fibonacci number ...
2
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1answer
125 views

Why is the sum of any ten consecutive Fibonacci numbers always divisible by $11$?

I was wondering if anyone has any insights regarding the fact that the sum of any $a_1, \dots, a_{10}$ consecutive Fibonacci numbers is divisible by $11$ (and furthermore equals to $a_7*11$). What can ...
3
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2answers
114 views

How do you determine if a number is a even Fibonacci number or not? [duplicate]

Rather than computing out the whole Fibonacci sequence and check if $n$ is even and in there, is there a more straightforward way to compute if $n$ is a even Fibonacci number?
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3answers
108 views

Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$

Given the Fibonacci, tribonacci, and tetranacci numbers, $$F_n = 0,1,1,2,3,5,8\dots$$ $$T_n = 0, 1, 1, 2, 4, 7, 13, 24,\dots$$ $$U_n = 0, 1, 1, 2, 4, 8, 15, 29, \dots$$ and so on, how do we show ...
1
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1answer
129 views

Proving formulas with products of Fibonacci numbers

While digging through my old notes, I stumbled upon some formulas involving multiplication of Fibonacci numbers that I discovered about 7 years ago (being fascinated with Fibonacci numbers at the ...
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2answers
269 views

proof by induction for golden ratio and fibonacci sequence

I have to prove the following equation by induction for $$x = \phi$$ I am stuck and I don't know how to proceed. This is the equation $$ \phi ^n = f_n\phi + f_{n-1} $$ where $f_n$ is the nth term ...
3
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1answer
67 views

Prove equality of Fibonacci sequence

Let $u(n)$ — the Fibonacci sequence. Prove that $$u(1)^3+...+u(n)^3=\frac{ 1 }{ 10 } \left[ u(3n+2)+(-1)^{n+1}6u(n-1)+5 \right].$$ I suppose we need prove that equality by iduction in $n$. ...
3
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1answer
76 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
2
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1answer
54 views

Fibonacci and Lucas numbers congruence relation?

The wikipedia page for Lucas Numbers seems to suggest that if $F_n ≥ 5$ is a Fibonacci number then no Lucas number is divisible by $F_n$. Here is the link. However, the page does not give any ...
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1answer
82 views

Is there a name for this Fibonacci Identity

Last night I was trying to solve a problem and discovered an identity relating to the Fibonacci sequence $$ \left\lvert F_{i-j}F_{i+j} - F_{i-k}F_{i+k} \right\lvert = \left\lvert F_{k - j}F_{k+j} ...
1
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1answer
74 views

Cycles in the Fibonacci Sequence mod n with matrices

I was just looking at this question about Fibonacci sequence cycles modulo 5, and I happened to see a very nice solution that involved using matrices. Using the matrix representation of the Fibonacci ...
0
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2answers
89 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. ...
4
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0answers
363 views

Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ...
0
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2answers
148 views

Alternating sum of product of Fibonacci numbers

Suppose that $\{F_n\}$ is the sequence of Fibonacci numbers. There is a well-known result that $$\sum_{i=1}^nF_i^2 F_{i+1}=\frac{1}{2}F_nF_{n+1}F_{n+2}.$$ This is easy to prove by induction. I was ...
4
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4answers
14k views

Induction proof on Fibonacci sequence: $F(n-1) \cdot F(n+1) - F(n)^2 = (-1)^n$

I can't seem to solve this problem. It is: The Fibonacci numbers $F(0), F(1), F(2),\dots $ are defined as follows: \begin{align} F(0) &::= 0 \\ F(1) &::= 1 \\ F(n) &::= F(n-1) + ...
6
votes
4answers
870 views

Last 10 digits of the billionth fibonacci number?

I want to compute the last ten digits of the billionth fibonacci number, but my notebook doesn't even have the power to calculate such big numbers, so I though of a very simple trick: The carry of ...
26
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3answers
1k views

Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?

$\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$ (Link), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ldots )$. This is ...
0
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0answers
57 views

Fibonacci series generated with division $1/999999999999999999999998999999999999999999999999$ [duplicate]

I tested the suggestion of finding the Fibonacci series by division, which sounded very surprising to me. I therefore used a simple sympy script to test it and found that it works as advertised. ...
2
votes
2answers
174 views

Question regarding the Fibonacci sequence

Given the Fibonacci sequence $(F_1, F_2,F_3, ...)$ how do I prove that if $m|n$ then $F_m|F_n$? Can this be proven with mathematical induction?
0
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2answers
63 views

Proving a Recursive Formula

I know there are some questions on this site about how to find a recursive formula, but I've already found the formula. I'm doing an assignment (http://mathstat.dal.ca/~svenjah/math2112/Assign6.pdf) ...
13
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6answers
322 views

Binomial Sum Related to Fibonacci: $\sum\binom{n-i}j\binom{n-j}i=F_{2n+1}$

How would I prove $$ \sum\limits_{\vphantom{\large A}i\,,\,j\ \geq\ 0}{n-i \choose j} {n-j \choose i} =F_{2n+1} $$ where $n$ is a nonnegative integer and $\{F_n\}_{n\ge 0}$ is a sequence of ...
2
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2answers
7k 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 ...