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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Identity on Fibonacci numbers: $F_{2n}^2=F_{2n+2}F_{2n-2}+1$?

Let $F_n$ be the Fibonacci Sequence ($F_1=F_2=1, F_{n+2}=F_{n+1}+F_{n}$). Prove that $F_{2n}^2=F_{2n+2}F_{2n-2}+1$. I've tried everything from induction to telescoping series but I haven't got close. ...
0
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0answers
127 views

A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
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2answers
39 views

Find $r$, given that $F_r= 2F_{101}+F_{100}$

Find $r$, given that $F_r= 2F_{101}+F_{100}$. We know that the recurrence relation for the Fibonacci sequence is $F_n= F_{n-1}+F_{n-2}$ and that $F_0 = F_1 = 1$, but how to proceed further?
4
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0answers
64 views

Finding first n so nth fibonacci is c modulo p

This is a question I stumbled upon in an online programming contest archive. The problem statement is simple, given $c \equiv F(n)$ mod $P$ and $P$, where $P$ is a prime of form 5$k$ + 1 or 5$k$ - 1, ...
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3answers
5k views

What will the recursion tree of Fibonacci series look like?

I am watching the Introduction to algorithm video, and the professor talks about finding a Fibonacci number in $\Theta(n)$ time at point 23.30 mins in the video. How is it $\Theta(n)$ time? Which ...
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2answers
41 views

Fibonacci numbers relation

I was wondering if there was a relation between a Fibonacci number and its position. Is there a function $f(n)$ such that $$f(n)=F_n$$ where $F_n$ is the nth Fibonacci number?
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2answers
64 views

Generating Functions with Fibonacci

a) Let \begin{align*} F_{\text{even}}(x) &= F_0x^0 + F_2x^2 + F_4x^4 + F_6x^6 + F_8x^8 + \cdots \\ &= x^2 + 3x^4 + 8x^6 + 21x^8 + \cdots \end{align*} be the generating function whose ...
3
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0answers
84 views

Multiple of $p$ in first $p+1$ Fibonacci Numbers

Defining $F_0 = F_1 = 1$ and $F_{n+1} = F_{n} + F_{n-1}$ for $n>0$ gives the Fibonacci sequence, and it is well-known that modulo $p$, one of the first $p+1$ terms is $0.$ In fact, more is known, ...
3
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1answer
331 views

Dominos ($2 \times 1$ on $2 \times n$ and on $3 \times 2n$)

How many ways are there to tile dominos (with size $2 × 1$) on a grid of $2 × n$? How about on a grid of $3 × 2n$?
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1answer
94 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
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2answers
76 views

Fibonnaci and Lucas series technique

Well I have the following two problems involving fibonnaci sequences and lucas numbers, I know that they share the same technique, but I don't have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: f_0 ...
1
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1answer
60 views

Proof Help dealing Lucas and Fibonacci Numbers

Claim: $L_n=F_{n-1}+F_{n+1}$ for all $n >0$ Could someone please help me prove this? My professor mentioned it in class, but didn't show us how to prove it. I am just curious. The $L$ stands for ...
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1answer
221 views

Solving a question about Fibonacci and Lucas numbers using induction

Im working on practice problems that the instructor gave us yesterday, and I absolutely have no clue of how to solve this problem.. I need to use mathmatical induction to solve this problem.. The ...
14
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4answers
334 views

How to prove $ \sum_{k=0}^n \frac{(-1)^{n+k}{n+k\choose n-k}}{2k+1}=\frac{-2\cos\left(\frac{2(n-1)\pi}{3}\right)}{2n+1}$

How to prove $$\sum_{k=0}^n \binom{n+k}{n-k}\frac{(-1)^{n+k}}{2k+1}=-\frac{2}{2n+1}\,\cos\left(\frac{2(n-1)\pi}{3}\right)\;\text{?}$$ I have a proof by induction for it, but it isn't simple! I want ...
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2answers
69 views

Why does Cassini's identity imply consecutive Fibonacci's number are relatively prime?

From Knuth's The Art of Computer Programming, Volume 1 on page 81 he gives Cassini's identity $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. Relation(4) and follows by saying "Relation(4) shows that $F_n$ ...
6
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1answer
112 views

Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
2
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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 $5x^2\pm~4$ ...
7
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1answer
300 views

Common terms in general Fibonacci sequences

Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For ...
2
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1answer
428 views

Lucas Numbers and Tilings

Show that $f_{n-1} + L_n = 2f_{n}$. So we need to find a $2$ to $1$ correspondence. Set 1: Tilings an $n$-board. Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet. So we need to ...
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6answers
291 views

show that $\frac{1}{F_{1}}+\frac{2}{F_{2}}+\cdots+\frac{n}{F_{n}}<13$

Let $F_{n}$ is Fibonacci number,ie.($F_{n}=F_{n-1}+F_{n-2},F_{1}=F_{2}=1$) show that $$\dfrac{1}{F_{1}}+\dfrac{2}{F_{2}}+\cdots+\dfrac{n}{F_{n}}<13$$ if we use Closed-form expression ...
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1answer
52 views

Converting Fibonacci number $F_{5n+3}$ to Lucas numbers $L_{n+k}$

I'm trying to prove that$F_{5n+3}\text{mod}10 = L_{n}\text{mod}10$. I rearranged it into a more solvable form of $F_{5n+3}-L_n = 10k$ (because if two numbers end in the same digit, their difference ...
2
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1answer
224 views

An identity involving Lucas numbers

Let $L_n$ be the Lucas numbers, defined by $L_n = F_{n-1} + F_{n+1}$ where f is the Fibonacci numbers. How to prove that $L_{2n+1} = \displaystyle \sum_{k=0}^{\lfloor n + 1/2\rfloor}\frac{2n+1}{2n+1 ...
3
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2answers
260 views

Fibonacci/Lucas Number Congruences

Is there a compendium of well-known (and elementary) Fibonacci/Lucas Number congruences? I've proven the following and would like to know if it is (a) trivial, (b) well-known, or (c) possibly new. $$ ...
1
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1answer
37 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.
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2answers
44 views

Express recursive function in Fibonacci

Given the Fibonacci function and the function $L_n = L_{n-1} + L_{n-2} + 1$, how do I go from this: $L_n + 1 = L_{n-1} + L_{n-1} + 1 + 1 \\ (L_n + 1) = (L_{n-1} + 1) + (L_{n-2} + 1)$ To this: $L_n = ...
47
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4answers
4k views

Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

So, $$1,1,2,3,5,8,13,21...$$ Any connection to primes?...it appears not. However, in between the Fibonacci numbers are how much primes? Let's see: 1 and 1 ZERO 1 and 2 NADA 2 and 3 ZILCH 3 and 5 ZIP ...
2
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1answer
30 views

Is this Bertrand's postulate-related statement valid?

Bertrand's postulate says: For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$. Is the following statement: For every $n>3$ there is always at least one ...
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2answers
77 views

How to solve the difference equation $u_n = u_{n-1} + u_{n-2}+1$

Given that: $$ \begin{equation} u_n=\begin{cases} 1, & \text{if $0\leq n\leq1$}\\ u_{n-1} + u_{n-2}+1, & \text{if $n>1$} \end{cases} \end{equation} $$ How do you solve this ...
4
votes
1answer
45 views

How to determine the generating function?

So I have $$\overset{*}{F} = \overset{*}{F}_{n-1} + \overset{*}{F}_{n-2} + g(n)$$ where $\overset{*}{F}$ is NOT a Fibonacci number for $n \geq 2$. $g(n)$ is any function $g: \mathbb{N} \to ...
4
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0answers
109 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 ...
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0answers
47 views

Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
4
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1answer
46 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
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1answer
33 views

Proving $F_n \ge (\frac{1}{2}(1+\sqrt{5}))^{n-2}$ for $n \in \mathbb{N}_{>1}$ when $F_n$ is the nth Fibonacci number

Let $F_n$ be defined as the nth Fibonacci number. Prove that $F_n \ge (\frac{1}{2}(1+\sqrt{5}))^{n-2}$ with $n \in \mathbb{N}_{>1}$ My approach thus far was to use induction over $n$. ...
2
votes
4answers
2k views

Proof the golden ratio with the limit of Fibonacci sequence [duplicate]

Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$ The exercise asks me to prove that: $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$. Sorry as can be proceed??
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1answer
39 views

An equation to prove with terms of Fibonacci sequence

I would like to prove an equation but I have stuck. The equation that is to prove is the below: $f(n)^2 + (-1)^{n+1} = f(n+1)f(n-1) , n \ge 2$. I'm trying to do an inductive proof of this equation. ...
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0answers
45 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...
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0answers
123 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\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 ...
2
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2answers
89 views

Proof by induction of a Fibonacci relation [duplicate]

We know: $F_0 = 0$ $F_1 = 1$ $F_i = F_{i-1} + F_{i-2}$ for $i \geq 2$ Prove by induction: $F_i = \dfrac{\phi^i-{\phi^{*}}^i}{\sqrt{5}}$ where $\phi = (1+\sqrt{5}) / 2$ and $\phi = (1-\sqrt{5}) / ...
4
votes
2answers
74 views

Conjecture: only one even Fibonacci term divided by two gives a prime: $F(9) = 34 = 2 \times 17$

Every Fibonacci term $F(3n)$ is divisible by two $F(3) = 2$ $F(6) = 8$ $F(9) = 34$ $...$ After seeking Fibonacci tables factorization until $F(10000)$, for every term $\frac{F(3n)}{2}$, it ...
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1answer
94 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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2answers
38 views

How does one arrive at a certain expression for the Fibonacci Zeta function?

In this paper by L. Navas, it is described how one can obtain a analytic continuation of the Fibonacci Dirichlet series (though I'm not sure it's actually a Dirichlet Series). First, the following ...
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0answers
121 views

Proving that a Fibonacci number is divisible by integer a

I am working on a review problem and can't figure out how to go about getting to an answer. We are told to let $F_n$ be the $nth$ Fibonacci number (defined as $F_1=F_2=1,F_{n+1}=F_n+F_{n-1}$). Show ...
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0answers
70 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
3
votes
1answer
68 views

Limit of A Fibonacci

Let $P(x)$ be an $n^{th}$ degree-polynomial which is defined below for some odd natural number $n$. And let us denote the set of roots of $P$ by $\{r_1,r_2,r_3,\dots, r_n \}$. $$\displaystyle P(x) = ...
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2answers
33 views

if d divides n then prove that fibonacci of d divides fibonacci of n

prove that if $d$ divides $n$ then prove that fibonacci of $d$ divides fibonacci of $n$. i have tried to write $F(n)$ as a multiple of $F(d)$ using the fact that $n = ad$ for some natural $a$ but got ...
7
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0answers
220 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 ...
2
votes
1answer
83 views

Covering a rectangle of size $n\times1$ with dominos

A rectangle of size $n\times1$ is given. (a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$? (b) In how many ways the rectangle can be covered with ...
2
votes
2answers
75 views

Proving a Problem involving Fibonacci numbers

I'm working on proving the problem that states $\text {The sequence}$ {$F_n$} $\text {is defined by the} \ F_1=F_2=1, F_{n+2}=F_{n+1}+F_n \ \text {for} \ n \ge 1.$ $\text {For any natural number m, ...
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4answers
120 views

Why the $GCD$ of any two consecutive fibonnaci numbers is $1$?

Note: I've noticed that this answer was given in another question, but I merely want to know if the way I'm using could also give me a proof. I did the following: $$F_n=F_{n-1}+F_{n-2} \\ ...
5
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4answers
4k views

How to prove gcd of consecutive Fibonacci numbers is 1? [duplicate]

Possible Duplicate: Prove that two any consecutive terms of Fibonacci sequence are relatively prime How to prove it ? Can you help me ? Let $f_n$ be Fibonacci Sequence. ...