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

4
votes
2answers
53 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
50 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 ...
1
vote
5answers
62 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
3answers
84 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
40 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?
1
vote
1answer
311 views

Prove by induction for $F(2n) = F(n)[F(n-1) + F(n+1)]$ for all $n\ge 1$

I am totally stumped by this question. I have proved the base case. Then for $k$ is $1$ assume the relation to be true. When I try to prove for $k+1$, the terms just do not simplify to what I want. Is ...
-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 $n>2$
5
votes
2answers
137 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.
2
votes
1answer
48 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
252 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{ ...
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) ...
9
votes
1answer
350 views

Infinite sum of reciprocal shifted Fibonacci numbers

I found on Wikipedia the following infinite sum : $$\sum_{k=0}^{\infty} \frac{1}{1+F_{2k+1}}=\frac{\sqrt{5}}{2}$$ There is no reference for this sum in the article and I couldn't find it ...
0
votes
2answers
78 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 $ ...
1
vote
3answers
81 views

Trying to prove $\sum_{i=1}^{N-2} F_i = F_N -2$

I'm trying to prove that $\sum_{i=1}^{N-2} F_i = F_N -2$. I was able to show the base case for when $N=3$ that it was true. Then for the inductive step I did: Assume $\sum_{i=1}^{N-2} F_i = F_N -2$ ...
8
votes
1answer
153 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
880 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 $n \in \mathbb{N}$?
1
vote
3answers
130 views

Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$

The Fibonacci sequence $F_0, F_1, F_2,\dots$ is defined by the rule $F_0=0, \ F_1=1, \ F_n = F_{n−1} + F_{n−2}$. Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$. So for the base case: ...
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.
-1
votes
1answer
44 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 ...
24
votes
1answer
601 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 ...
3
votes
1answer
248 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 = ...
4
votes
0answers
128 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 ...
11
votes
1answer
236 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 ...
2
votes
1answer
133 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 ...
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 ...
5
votes
1answer
192 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
51 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) ...
-1
votes
1answer
40 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
votes
1answer
75 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 ...
0
votes
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
votes
1answer
94 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)} ...
6
votes
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 ...
2
votes
2answers
109 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
votes
2answers
92 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
votes
2answers
84 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
votes
1answer
63 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
votes
2answers
202 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 ...
5
votes
0answers
87 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} ...
1
vote
6answers
185 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 ...
2
votes
1answer
162 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
votes
2answers
118 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?
27
votes
2answers
820 views

Fibonacci $\equiv -1 \mod p^2$

Is there a prime $p > 3$ such that the Fibonacci number $F_{np} \equiv -1 \mod p^2$ for some natural number $n$? I know none of the first $1000$ primes $> 3$ qualify. EDIT: In response to ...
6
votes
3answers
111 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 ...
3
votes
1answer
79 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
votes
1answer
58 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 ...
1
vote
0answers
77 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 ...