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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3
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4answers
287 views

Fibonacci sequence divisible by 3?

I have a recursion question for my combinatorial class. I'm looking at the Fibonacci sequence $f(n)=f(n-1)+f(n-2)$ for $n \geq 3$ with $f(1)=f(2)=1$. I'm trying to prove that $f(n)$ is divisible by 3 ...
1
vote
2answers
80 views

Source and/or combinatorial interpretation for $F_{n+k} = \sum_{i=0}^{k} \binom{k}{i}F_{n-i}$

Through some fussing with Taylor's Theorem in the discrete calculus described here (among other places), I found what I believe to be an identity: $$F_{n+k} = \sum_{i=0}^{k} \binom{k}{i}F_{n-i}$$ ...
7
votes
1answer
84 views

Is there a non-constant function $f$ such that $f'(x) = f(x - 1)$?

In discrete calculus, where the difference operator $\Delta f = f(x + 1) - f(x)$ takes the place of $\frac{d}{dx}$, Fibonacci sequences are given by the functions satisfying: $$ \Delta f(x) = f(x - ...
0
votes
1answer
55 views

Sum of $i$ times $(i-1)^\text{th}$ Fibonacci Number [closed]

Consider the expression $$\sum\limits_{i=1}^n i \cdot F_{i-1}$$, where $F_{0}=0, F_{1}=1, F_{2}=1, F{3}=2,$ etc. Is there a closed formula for this? If so, find it.
0
votes
0answers
37 views

Finding the n-th Pisano Period (for small n)

From Wikipedia: [...] the $n$th Pisano period, written $\pi(n)$, is the period with which the sequence of Fibonacci numbers taken modulo $n$ repeats. For example, the Fibonacci numbers modulo 3 ...
7
votes
3answers
159 views

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$.

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$. Assume $\binom{n}{k} = 0$ if $k>n$. Does anyone know an elementary ...
0
votes
2answers
33 views

Proving an identity of Fibonacci Numbers by induction

Say we know this as a given: $$E_0 = A$$ $$E_1 = B$$ $$E_2 = A + B$$ $$E_3 = A + 2B$$ $$E_4 = 2A + 3B$$ $$E_5 = 3A + 5B$$ $E_{n+1}$ is defined as: $$E_{n+1} = E_n + E_{n-1}$$ You can start to see ...
0
votes
1answer
48 views

Prove that for all $n \in Z^+$, Euclid’s algorithm applied to the pair $(f_{n+1}, f_{n+2})$, takes $n$ steps.

Prove that for all $n \in Z^+$, Euclid’s algorithm applied to the pair $(f_{n+1}, f_{n+2})$, takes $n$ steps. (Here, as previously, $f_n$ denotes the nth Fibonacci number.) I don't understand how to ...
0
votes
1answer
70 views

Prove that if Euclid's algorithm applied to the pair $(u,v)$ takes $n$ steps, then $u \geq f_{n+2}$ and $v \geq f_{n+1}$

Let $u, v \in Z^+$ satisfy $u > v$. Prove that if Euclid's algorithm applied to the pair $(u,v)$ takes $n$ steps, then $u \geq f_{n+2}$ and $v \geq f_{n+1}$. Where the $f_n$ values refer to the ...
1
vote
0answers
34 views

Periods of Fibonacci numbers in mod. Number Theory

Work out the periods $π(n)$ of the $\mod n$ such that $$f_k ≡ f_{k+π(n)} \mod n$$ I got $π(2)=3$,$π(3)=8$,$π(4)=6$ by computing it and looking at the periods. Now Part 2 Prove that for all ...
2
votes
2answers
50 views

Using induction to prove a formula for the Fibonacci sequence involving the solutions of $x^2=x+1$

Let $\{f(n)\}_{n=1}^{\infty}$ denote the Fibonacci sequence defined by $f(1)=1, f(2)=1$, and $f(n)=f(n-1)+ f(n-2)$ for all $n\geq 3$. Let $α=\dfrac{1+\sqrt{5}}{2}$ and $β=\dfrac{1-\sqrt{5}}{2}.$ ...
0
votes
1answer
70 views

Mathematical induction proof of $\sum_{i = 1}^{n} F_{2i} = F_{2n + 1} - 1$

Use Mathematical Induction to show that $$\sum\limits_{i=1}^n F_{2i}=F_{2n+1}-1$$ for all integer $n\geq1$. My answer: Base case: for n = 1 $$\sum_{i = 1}^{n} F_{2i} = \sum_{i = 1}^{1} F_{2i} = ...
0
votes
1answer
42 views

Proving Fibonacci sequence with mathematical induction

Okay, so I have the following thing: $$\sum_{i=1}^a F_{2i}=F_{2a+1}-1 $$ It's to do with Fibonacci sequence. I can do the basis step of MI fine (proving for $a = 1$) However the inductive step has ...
0
votes
3answers
83 views

Proof on Fibonacci sequence: $F(1) + F(3) + \cdots + F(2n-1) = F(2n)$ using induction and recursion

The problem is: Use induction and the recursive formula to prove that: $$F(1) + F(3) + \cdots + F(2n-1) = F(2n)$$ For the base case I let $n=1$ which gave $$F(1) = F(2(1))$$ $$1=1$$ Which is ...
0
votes
0answers
59 views

Does the p-adic valuation of n, apply to Fibonacci numbers and Fibonacci-Wieferich primes?

Does the p-adic valuation of n, apply to Fibonacci-Wieferich primes in the following way? Let Fup be the smallest Fibonacci number divisible by a prime p > 5, then Vp(Fupk) = Vp(Fup) + Vp(k). The ...
3
votes
3answers
204 views

Combinatorics — Fibonacci: formula for $F_1+F_3+ \cdots +F_{2n+1} $

For the following expression, find a simple formula which only involves one Fibonacci number. Then prove it by induction. $$F_1+F_3+ \cdots +F_{2n+1} $$ I'm be appreciated for any help. I have no ...
1
vote
5answers
63 views

Proof about specific sum of Fibonacci numbers

Let $F_k$ denote the $k$-th Fibonacci number. Find a formula for and prove by induction that your formula is correct for all $n > 0$. $$ (-1)^0 F_0+(-1)^1 F_1+(-1)^2 F_2+\cdots+(-1)^n F_n=\ ? $$ I ...
2
votes
1answer
86 views

How to find pythagoras triplet using the fibonacci sequence?

I'm using the Fibonacci sequence to generate some Pythagorean triples $(3, 4, 5,$ etc$)$ based off this page:Formulas for generating Pythagorean triples starting at "Generalized Fibonacci Sequence". ...
0
votes
1answer
31 views

Fibonacci Sequence Squared

I have been learning about the Fibonacci Numbers and I have been given the task to research on it. I have been assigned to decribe the relationship between the photo (attached below). I know that the ...
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 ...
0
votes
1answer
76 views

Consider a recursive sequence. Find all values of x for which this sequence is bounded

Consider a recursive sequence $a_{n+1} = a_n + a_{n-1}$ for all $n \geq 2$ with $a_{1} = 1$ and $a_2 = x$. Find all values of $x$ for which this sequence is bounded.
4
votes
2answers
55 views

Calculating number of tile sequences

My daughter (aged 12) came to me with the problem below. I was able to help her to some extent but I could not see an age-appropriate solution. That is, I could imagine solutions involving factorials ...
1
vote
1answer
52 views

Are Fibonacci numbers with a square prime index always divisible by $F_p$?

I am doing some research on sequences and I need some help. The sequence of $F_{p^2}$ seems sort of different. It seems that because the index only has one distinct prime factor, as a result the only ...
1
vote
1answer
59 views

Showing $P_n = {F_n \over {2^n}}$

Let $P_n$ be the probability that, if you flip a fair coin $n$ times, there are no consecutive heads. Also, let $F_n$ be the $n^{th}$ Fibonacci number, normalized by $F_1 = 1$ and $F_2 = 2$ and $F_n = ...
1
vote
1answer
75 views

A Property of Fibonacci Numbers [duplicate]

I've seen the property $$f_{n+1} f_{n−1} = f_n^2 + (−1)^n, n ≥ 2.$$ for Fibonacci numbers at Abstract Algebra book of Thomas W. Judson. I've tried it for a few Fibonacci number, and I've really ...
1
vote
1answer
36 views

Fibonacci sequence developing [duplicate]

For the sum $$\sum_i^n {n-i \choose i}$$ I evaluate it for $n=1,2,3,4,5$ For $n=1$ we have $$\sum_{i=0}^1 {1-i \choose i} = {1 \choose 0} + {0 \choose 1} = 1 + 0 = 1$$ For $n=2$ we have ...
3
votes
1answer
49 views

Showing $F_{\frac{p^2+1}{2}}\equiv p-1 \pmod{p}$ when $p\equiv \pm 2 \pmod{5}$ and $p\equiv 3 \pmod{4}$

A while back I was messing around with representations of finite fields and found this problem above while doing so. I'll explain below how I came to this point but my question is: Question: How ...
3
votes
1answer
43 views

Hankel determinant involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number, with $F_1 = F_2 = 1$. Denote by $M\left(n\right)$ the $n \times n$ Hankel matrix with $\left(i,j\right)$-th entry $F_{i+j-1}^{n-1}$, where $i$ and $j$ ...
1
vote
0answers
78 views

Proving that the g.c.d of non-consecutive Fibonacci numbers is also a Fibonacci number

I'm trying to prove that: $\gcd\left(u_n, u_m\right) = u_{\gcd\left(n,m\right)}$ for any positive integers $n$ and $m$. Here, $u_n$ denotes the $n$-th Fibonacci number. I know that consecutive ...
13
votes
2answers
270 views

An analogue of Hensel's lifting for Fibonacci numbers

Let $F_0, F_1, F_2, \ldots$ be the Fibonacci numbers, defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n\geq 2$. In this question Oleg567 conjectured the following interesting ...
2
votes
3answers
88 views

Adding subscripts

This is a stupid question. But I'm trying to solve a Fibonacci problem and just realized that I don't know how to manipulate them. For example why does $F_{3n+1}$=$F_{3n-1}$+$F_{3n}$
0
votes
0answers
41 views

Fibonacci and Lucas congruence

I'd like to prove this congruence: $F_{2kt+n} \equiv (-1)^t F_n \pmod{L_k}$, where $F_n$ and $L_n$ are the Fibonacci and Lucas sequences. I have no idea how can i start. May anyone help?
5
votes
1answer
196 views

conjectured new generating function of fibonacci numbers

I conjecture a new generating function for the fibonacci numbers $F_{n}$. Given,the following conjectured q-continued fraction ...
-2
votes
1answer
45 views

Inequality involving Fibonacci numbers [closed]

If $F(n)$ are Fibonacci's numbers then prove that $$1< \frac{F(n+1)}{F(n)}<2$$ for all $n>2$
2
votes
1answer
49 views

A sequence related to squares of Fibonacci nubers

Let $f(n)$ be defined by $f(n)=f(n-1)+f(n-3)+f(n-4)$, for $n \ge 5$, $f(1)=1, f(2)=1, f(3)=2, f(4)=4$. First few terms of the sequence $(f(1), f(2), f(3), \ldots$) look like $(1, 1, 2, 4, 6, 9, ...
2
votes
2answers
266 views

Fibonacci sequence and eigenvalues

I'm learning about eigenvectors and values, and one of the excercises in my book tackles the fibonacci recursion from this angle. Let $F = \begin{bmatrix}1&1\\1&0\end{bmatrix}^n \quad\text{ ...
5
votes
2answers
140 views

On the Fibonacci sequence: is there an infinite number of primes $p$ dividing $F_{p-1}$?

Let $\{F_n\}_{n\geq 0}$ be the Fibonacci sequence. Prove that the number of primes $p$ so that $p\mid F_{p-1}$ is infinite. I tried to use induction, to no avail.
34
votes
3answers
2k views

Fibonacci infinite sum resulting in $\pi$

I found the following identity. While trying to prove it, I found some things that I don’t quite understand: $$\frac{\pi}{4}=\sqrt{5} \sum_{n=0}^{\infty} \frac{(-1)^n F_{2n+1}}{(2n+1) ...
0
votes
2answers
79 views

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
votes
0answers
131 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
votes
1answer
158 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 ...
16
votes
5answers
885 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
votes
1answer
610 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
votes
1answer
47 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
votes
0answers
34 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.
3
votes
1answer
365 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
votes
1answer
178 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
votes
1answer
237 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
votes
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
votes
4answers
99 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 ...