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 $F_1=1$.

learn more… | top users | synonyms

1
vote
3answers
374 views

How to compute the nth number of a general Fibonacci sequence with matrix multiplication?

If we want to compute the nth Fibonacci number we just power the matrix $M = \left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right]$ $n$ times and we get $M =\left[ \begin{array}{cc} ...
2
votes
1answer
118 views

The number of partitions of $n$ and the $n$th Fibonacci number.

I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let $P_n$ be the number of partitions of $n$ ...
8
votes
2answers
304 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
1
vote
1answer
68 views

Fibonacci numbers identity - proof by induction

$\displaystyle F_{k-1} F_{k+1} - F_k^2 = (-1)^k$ I have done the base step for $k=1$ and it works. I realize we need to prove for $k+1$, so: $$F_k F_{k+2} - F_{k+1}^2 = (-1)^{k+1}$$ Could ...
16
votes
4answers
1k views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
1
vote
1answer
78 views

Fibonacci numbers $F_{n+3} + F_{n} = 2F_{n+2}$

Prove $F_{n+3} + F_{n} = 2F_{n+2}$ for any positive integer n. So What I did was this: fn+ fn+1 = fn+2 fn + fn+1 = fn+2 => fn+2 -fn+1 fn+1 + fn+2 = fn+3 then I subsituted into equation in ...
1
vote
4answers
175 views

Fibonacci Calculation using a larger matrix

So the formula to generate the fibonacci sequence in matrix form is: $$ \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & ...
11
votes
5answers
2k views

How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or ...
0
votes
1answer
1k views

Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
1
vote
2answers
81 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
0
votes
1answer
430 views

Asymptotic value of Fibonacci numbers

It is well known that $F_n\sim\frac{\phi^n}{\sqrt{5}}$, where $\phi=\frac{1+\sqrt{5}}{2}$. Does someone know a better estimate? With proof please. I'm trying to calculate the following limit: Let ...
0
votes
1answer
83 views

Let $u_n$ be the $n$-th entry in the Fibonacci sequence $1,1,2,3,5,8,13,\ldots$

If you start with $u_1 = 1$ and $u_2 = 1$, then the sequence can be generated using the formula $$u_{n+1} = u_n + u_{n-1}\ .$$ If $u_n = r^n$, what is r? Can anyone figure this out? I am so stuck ...
3
votes
2answers
651 views

The sum of $n$ consecutive Fibonacci numbers.

The sum of $8$ consecutive Fibonacci numbers is divisible by $3$. How can I generalize this for the sum of $n$ consecutive Fibonacci numbers? For example, $$1+1+2+3+5+8+13+21=54=3\times 18 \\ ...
2
votes
1answer
109 views

Proof that golden angle successively divides the largest gap by the golden ratio?

The golden angle divides the circumference of a circle by the golden ratio. "If radial spokes are placed successively into the circle, each spaced by a golden angle increment, then each additional ...
4
votes
1answer
95 views

What's the Lucas version of the Möbius test for Fibonacci numbers?

I recently came across the following, attributed to Möbius: $$(a\in\mathbb N)=F_n\iff\left[\varphi a-\tfrac{1}{a},\varphi a+\tfrac{1}{a}\right]\ni(b\in\mathbb N)$$ It is the lesser-known test used to ...
2
votes
2answers
110 views

How do I apply the $\pm4$ part of the equation $5F_n^2\pm~4=L_n^2$ without knowing $n$?

I'm trying to test a great many numbers $a^3+b^3$ to see if any of them are Fibonacci using the formula $$a^3+b^3=F_n \iff 5(a^3+b^3)^2\pm~4=L_n^2$$ I want to make my search more efficient by having ...
0
votes
1answer
61 views

Looking for $\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$

I am looking for the sum of the series: $$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$$ where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the ...
1
vote
1answer
165 views

What is length of period of fibonacci number mod 1000033

Can some explain me how the period of Fibonacci mod $1000033$ is $4684$. As we know if $n$ mod $5$ is $2$ or $3$ then period is $2n + 2$ so the period should me $2\times1000033 + 2$ but why it is ...
2
votes
1answer
49 views

finding $n$ consecutive composite Fibonacci numbers.

For each $n$, How can we find $n$ consecutive composite Fibonacci numbers?
1
vote
2answers
61 views

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers. Progress: $F_{11}=89$ . I believe you should find the period of $F_n \bmod 89$ and use that to solve it. But I'm not not ...
1
vote
4answers
723 views

Determine which Fibonacci numbers are even

(a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture. (b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical ...
2
votes
3answers
98 views

Is it true that $5^k \mid f(5^k)$?

I guess if it is true that $5^k \mid f(5^k)$, where $f(n)$ denotes the $n$-th Fibonacci's number. I have tried to prove it by induction on $k$, but nothing. Have you got any ideas?
2
votes
1answer
82 views

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod 1001$, where $F_n$ denote the Fibonacci numbers. I have tried using the fact that $F_{n^k} \bmod F_n = 0, k=1,2,3,...$ but that doesn't get me anywhere. Thanks!
3
votes
0answers
60 views

Are $1$ and $3$ the only numbers of the form $2^n-1$ that exist in the Fibonacci sequence? [duplicate]

Are $1$ and $3$ the only numbers of the form $2^n-1$ that exist in the Fibonacci sequence? Then, if they are not the only ones, are there infinite cases? I have tried finding another example using a ...
1
vote
1answer
152 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
2
votes
1answer
57 views

What is the Lucas counterpart to the Fibonacci identity $5F_n^2\pm~4=\lambda^2$?

It's a well-known rule that a number $x$ belongs to the Fibonnaci Sequence iff: $$\begin{align}5x^2\pm~4&=\lambda^2&\lambda\in\mathbb Z\end{align}$$ In other words, if and only if $5x^2\pm~4$ ...
0
votes
2answers
75 views

Help with a proof involving Fibonacci numbers.

I'm working through SICP MIT course, and I'm a little lost on how to prove the following statement. I think I'm able to demonstrate it, but have no idea how to prove this statement. I may ...
13
votes
6answers
331 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
votes
0answers
158 views

Fibonacci applied to human population living to dead ratio problem

If this forum is not the right one for my question, please redirect it. I do not know where to ask it. The question might seem tongue-in-cheek, believe me it's not! Last week to occupy my mind, I ...
1
vote
2answers
217 views

Non integer Fibonacci numbers

I'm pretty sure we're all familiar with the Fibonacci sequence. Most people with more than passing knowledge of this most marvelous gem are aware of the Binet formula, $Fib(n) = (\varphi^n - ...
5
votes
1answer
133 views

Number of zeros in Fibonacci sequences mod $p$

We know that Fibonacci sequences are periodic in mod $m$. For example, for $p\equiv \pm 1 \pmod 5$ and $\pm 2 \pmod 5$ the periods for Fibonacci sequences modulo $p$ divide $p-1$ and $2p+2$ ...
1
vote
2answers
58 views

Help with a proof I can't quite

Let $(F_j)^\infty_{j=1}$ be the sequence of Fibonacci numbers. For all $n \in \mathbb{N}$, $\sum\limits_{k=1}^{2n-1}F_kF_{k+1}=(F_{2n})^2$. I handled the base case quite well but couldn't go very ...
1
vote
1answer
38 views

Lucas Numbers Inequality

Can it be shown that \begin{align} \frac{1}{\ln(1+L_{n}) -1} \geq \frac{L_{n}}{(L_{n}-1)(e^{L_{n}}-1)} \end{align} where $L_{n}$ is the $n^{th}$ Lucas number. Show results in full detail.
1
vote
0answers
168 views

Linear combination of numbers: Express the $n^{th}$ Fibonacci number in terms of known constants.

Given the concept that any number can be expressed as a combination of other numbers can the $n^{th}$ Fibonacci number be expressed in terms of $\zeta(3)$ and $\ln(2/e)$ ? If possible, show all work ...
33
votes
4answers
1k views

Conjecture: Only one Fibonacci number is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below ...
3
votes
1answer
119 views

Using matrices to calculate fibonacci?

I have been told a couple of times it possible to calculate the fibonacci sequence much quicker using matrices but I never understood/they never elaborated. Would somebody be able to show how this ...
3
votes
2answers
295 views

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?
1
vote
4answers
238 views

Summation of Fibonacci numbers $F_n$ with $n$ odd vs. even

Compare the summation below: $$\begin{align} \smash[b]{\sum_{i=1}^n F_{2i-1}}&=F_1+F_3+F_5+\cdots+F_{2n-1}\\ &=1+2+5+\cdots+F_{2n-1}\\ &=F_{2n}\\ \end{align} $$ with this one: ...
3
votes
2answers
190 views

Check if the number is a result of multiplying two fibonacci numbers

Is there a way to check if a given number is a result of multiplication of two fibonacci numbers? Any theory/characteristics allowing to quickly decline some of these given numbers?
1
vote
1answer
153 views

Does this mean some Wall-Sun-Sun primes have already been found?

In the PrimeGrid project statistics page for Wall-Sun-Sun Prime Search, it says, Wall-Sun-Suns ... 2 Near Wall-Sun-Suns ... 208 However, all the internet search ...
8
votes
2answers
6k views

How to prove Fibonacci sequence with matrices? [duplicate]

How do you prove that: $$ \begin{pmatrix} 1 & 1\\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n\\ F_{n} & F_{n-1} \end{pmatrix}$$
2
votes
3answers
586 views

Induction proof $F(n)^2 = F(n-1)F(n+1)+(-1)^{n-1}$ for n $\ge$ 2 where n is the Fibonacci sequence [duplicate]

Prove that $F{_n}^2 = F_{n-1}F_{n+1}+(-1)^{n-1}$ for n $\ge$ 2 where n is the Fibonacci sequence F(2)=1, F(3)=2, F(4)=3, F(5)=5, F(6)=8 and so on. Initial case n = 2: $$F(2)=1*2+-1=1$$ It is true. ...
0
votes
1answer
70 views

Proofs Related to the Fibonacci Sequence

I need to prove several proofs related to the Fibonacci sequence and I don't have the faintest clue how to do so. Please help! Given that the Fibonacci sequence is defined as $f_n = f_{n-1} + ...
34
votes
4answers
727 views

Fibonacci numbers from $998999$

Is there a nice explanation of ...
1
vote
3answers
133 views

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 $3$ ...
1
vote
2answers
51 views

Relation between series and equations

There is following quotes from wiki on Plastic number: The powers of the plastic number $A(n) = ρ^n$ satisfy the recurrence relation $A(n) = A(n − 2) + A(n − 3)$ for $n > 2$. And 2nd is that ...
4
votes
0answers
114 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on Fibonacci quadratic residue: $F(x)^2 mod F(y)$ = { if y is even: $(F(|x-y|)^2$ } { if y is odd: $-(F(|x-y|)^2$ } for ...
0
votes
1answer
67 views

Factoring for Strong Induction for Fibonacci Sequence

Fibonacci: prove the following theorem: define the Fibonacci sequence $\left\{ a_n\right\}_{n=0}^{\infty}$ by $a_0=a_1=1$ and for integers $k>1$, $a_k=a_{k-1}+a_{k-2}$. Then, for each integer $n$, ...
4
votes
2answers
195 views

Successively longer sums of consecutive Fibonacci numbers: pattern?

Consider the following: $$\begin{align} F_{n-1}+F_{n-2}&=F_n\\ F_{n-1}+F_{n-2}+F_{n-3}&=F_{n-1}+F_{n-1}\\ &=2F_{n-1}\\ F_{n-1}+F_{n-2}+F_{n-3}+F_{n-4}&=F_n+F_{n-2}\\ &=L_{n-1}\\ ...
1
vote
4answers
50 views

Help understanding Recursive algorithm question

We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$ For $n\geq0$, let $c(n)$ be the total number of additions for calculating $f(n)$ ...