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

0
votes
0answers
25 views

Prove that for any power function $f_n = c^{n}$, the number of decimal digits of $ f_{10^n}$ is given by $10^{n}log_{10}c$

I am reading this page about some interesting properties of the Fibonacci numbers: http://mathworld.wolfram.com/FibonacciNumber.html The following is said: The numbers of Fibonacci numbers less ...
0
votes
2answers
44 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. O(n^2) c. O(2^n) d. ...
0
votes
0answers
45 views

Identify fibonacci sequences from a set of data

Let there be a set of increasing order integer data ${a_1, a_2, a_3, a_4, ...}$. given the increasing infinite sequence of integers, how can we determine whether there is an infinite subsequence which ...
0
votes
0answers
32 views

Fibonacci polynomial summation

The $n^{th}$ value of a polynomial ($S_n$) of order $k$ is a polynomial in $x$ is given by : $\left(\frac{S_n}{x^n}\right)$= $\sum_{j=0}^n \left(\frac{{F_j}^k}{x^j}\right)$ Where ${F_j}$ is the ...
0
votes
1answer
21 views

Show that $\forall n \in \mathbb{N}: n$ can be written as $F_{n+1}$ different sums of ones and twos

Show that $\forall n \in \mathbb{N}: n$ can be written as $F_{n+1}$ different sums of ones and twos where the order matter. Presumably, mathematical induction can be leveraged here. Step 1: Show ...
0
votes
2answers
41 views

Applying the mean value theorem to the closed form of the Fibonacci sequence?

Is it possible to apply the mean value theorem to the closed form of the Fibonacci sequence for the 7 numbers starting at 1 and ending with 13 (inclusive)? It's been a LONG time since I studied ...
0
votes
1answer
47 views

Deduce a series formula for product of Fibonacci numbers.

Start with the arbitrary pair of Fibonacci numbers $F_{n+1}$, $F_n$ and apply the Euclidean Algorithm to it. Deduce a series formula for the product $F_{n+1}F_n$. I use the formula, $F_{n+1} = F_n + ...
0
votes
1answer
58 views

Proving a function by induction [duplicate]

Let $f(n)$ be the function defined by $$ f(n) = \frac 1{\sqrt{5}} \left[ \left(1+\sqrt{5}\over2\right)^n- \left(1-\sqrt{5}\over2\right)^n \right] $$ How do you prove that $f(n) = f(n+2) - f(n+1)$ ...
2
votes
4answers
96 views

Relationship between Fibonacci's secuence and $x^2 - x - 1$.

On the end of Apostol's Mathematical Analysis' first chapter, one can find the following exercise (and I paraphrase): Prove that the $n$-th term of the Fibonacci sequence is given by $$x_n = ...
0
votes
2answers
38 views

Fibonacci even numbers formula

i found a general formula in any given set of Fibonacci numbers ,to find the next given even number we can use the formula E*4 + Eo where E is the given even number Eo is the even number ...
0
votes
3answers
86 views

Lucas numbers and fibonacci

This is a question straight from the Applied Combinatorics book. Suppose that chairs are arranged in a circle. Let $L_n$ count the number of subsets of $n$ chairs which don't contain consecutive ...
4
votes
0answers
88 views

Are there infinite Fibonacci primes if and only if there are infinite Fibonacci numbers that are Fibonacci pseudoprimes?

One of the open questions about the Fibonacci numbers is if there are infinite prime numbers inside the Fibonacci sequence. I wonder if a good approach would be trying to know first if there are ...
2
votes
2answers
34 views

Prove the sum of the even Fibonacci numbers

Let $f_n$ denote the $nth$ Fibonacci number. Prove that $f_2\:+\:f_4\:+...+f_{2n}=f_{2n+1}-1$ I am having trouble proving this. I thought to use induction as well as Binet's formula where, ...
2
votes
0answers
160 views

LCM of Fibonacci numbers

$\newcommand{\lcm}{\operatorname{lcm}}$There is a nice property of Fibonacci numbers which says that: $$\gcd(F_{a_1}, \ldots, F_{a_n}) = F_{\gcd(a_1, \ldots, a_n)}$$ I am curious is there anything ...
3
votes
1answer
52 views

Showing that $f_n$ is the number closest to $\frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n$ [duplicate]

I want to prove that the the $n$th Fibonacci number $f_n$ is the integer closest to $\frac{1}{\sqrt{5}}(\frac{1+\sqrt{5}}{2})^n$. What would be a rigorous way to go about this? I assume I'll have to ...
1
vote
0answers
22 views

Hankel determinant involving Fibonacci numbers

Let $F_n$ denote the nth Fibonacci number, with $F_1 = F_2 = 1$. Denote by M(n) the nxn Hankel matrix with $i,j $ entry $F_{i+j-1}^{n-1}$, where i and j range from 1 through n. Finally, let d(n) = ...
5
votes
1answer
77 views

Connection between Euler's totient function and Fibonacci numbers

For a sequence $(a_n)$ of natural numbers define $\alpha(n):=\min\{m\in\mathbb{N}:n|a_m\}$ whenever it exists. Thus $\alpha(n)$ is the first index $m$ such that $n$ divides $a_m$. Now define the ...
0
votes
1answer
37 views

Convergence of fibonacci quotient $\frac{f_n}{f_{n+1}}$

I know that $\frac{f_{n+1}}{f_{n}}$ converges against $\phi = \frac{1+\sqrt{5}}{2}$. The question i want to to ask you is if the following conclusion is correct, I mean i know that if we have two ...
-1
votes
1answer
43 views

What is the probability that a given a positive number, will be found in the space of shifted fibonacci sequences?

We are given a space of shifted Fibonacci sequences, Fk, Fk+1, Fk+2, Fk+3, Fk+6, Fk+8, Fk+9, Fk+10, Fk+10, Fk+11….. Given a number,n, what is the probability of this number within this space? And ...
6
votes
0answers
162 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 ...
3
votes
3answers
217 views

How many times a positive number can be found in shifted Fibonacci Sequences?

Given a positive number, how many times can this number be found in shifted fibonacci sequences? ...For example...Number 11, can be present seven times in total, in Fn+3, Fn+6, Fn+8, Fn+9, Fn+10, ...
3
votes
2answers
68 views

How to find a Fibonacci number that is divisible by $x$?

I'm looking for an algorithm that is better than just checking every number in the Fib Sequence for divisibility. Example: Find the first Fib number that is divisible by $x=223321$, with no ...
0
votes
2answers
95 views

Find a formula for the nth Fibonacci Number [duplicate]

So I'm being asked to find a formula for the nth fibonacci number. I know the answer is $$x_{n}=\frac{(1+5^{1/2})^{n} -(1-5^{1/2})^n}{\sqrt{5}2^n}$$ However I don't really know how to get there. ...
1
vote
2answers
49 views

Seeking combinatorial proof for $F_{n+1} -1=\sum\limits_{k=0}^{n-1} F_k$

In order to give a combinatorial proof for this equation, we need to find what these two count for. But I don't know what they count for and how I can pivot the RHS to show that it actually counts ...
5
votes
2answers
92 views

Prove function is Fibonacci sequence

Stuck on how to finish a question I'm working on. Have to find the number of bitstrings of length n with no odd length maximal runs of ones. For example, when n=3 there are three such bitsrings: 011, ...
1
vote
1answer
44 views

Lucas Number Sequence

Can anyone help me in this question: Define $ (b_n)$ as $b_1= 1,b_n=a_{n+1} - a_n $ for $ n\ge 2$, where $ a_n $ is the Fibonnaci series. This sequence is known as the sequence of Lucas numbers. ...
7
votes
3answers
168 views

Show that there are infinitely many integers such that $ \binom{m}{n-1} = \binom{m-1}{n} $

This question comes from the 1st Brazilian's IMO TST of 2004. I have found no solutions of it online, though I have developed one. After getting to $ mn = (m-n)(m-n+1) $, my solution relies on the ...
1
vote
1answer
36 views

Fibonacci Numbers, show $F_n$ $>=$ $2^{n/2}$ for n $>=$ 6.

I want to show that for the fibonacci numbers, $F_n$ $>=$ $2^{n/2}$ for n $>=$ 6. My thought was to prove this via induction. I showed the base case is true for $F_n$, n=6 and 7. I assumed ...
8
votes
2answers
305 views

If $ab-1$, $bc-1$, $ca-1$, $ab-a-b+c$, $bc-b-c+a$, $ca-c-a+b$ are perfect squares, then are $ab+a+b-c$, $bc+b+c-a$, $ca+c+a-b$ also perfect squares?

About a month ago, a friend of mine taught me that there exist many sets of three positive integers $(a,b,c)$ where $a\not=b,b\not=c$ and $c\not=a$ such that each of $$ab-1,\ bc-1,\ ca-1,\ ab-a-b+c,\ ...
1
vote
1answer
50 views

Proving the Fibonacci identity $\sum_{i=1}^n f_i^2=f_nf_{n+1}$ by induction [duplicate]

I am having troubles with a proof question. Prove that for any $n\ge1$, $\sum_{i=1}^n f_i^2=f_nf_{n+1}$, where $f_n$ is the $n$'th Fibonacci number. I have the base case and the induction ...
4
votes
2answers
86 views

Of fibonomials, pellonomials, and tribonomials, etc

I. Linear recurrence with order 2 Given the Fibonacci numbers $F_n$, we have $$\begin{aligned} &F_n+F_{n+1}-F_{n+2}=0\\[1mm] &F_n^2-2F_{n+1}^2-2F_{n+2}^2+F_{n+3}^2=0\\[1mm] ...
2
votes
0answers
54 views

Generalizing the Fibonacci identity $F_{2n}=-F_{n-1}^2+F_{n+1}^2$

Using an integer relations algorithm, we get, $$F_{2n}=-F_{n-1}^2+F_{n+1}^2$$ $$6F_{4n}= -F_{n-2}^4-3F_{n-1}^4+3F_{n+1}^4+F_{n+2}^4$$ The pattern of the subscripts is clear. Expressing the ...
1
vote
2answers
62 views

formula for logarithmic spiral on a linear level

I am trying to plot the contents of a circle, which include geometric elements and spirals, on a linear graph. For example, take a circle, take the beginning and the end and make it straight. What ...
3
votes
2answers
52 views

Fibonacci recursive algorithm yields interesting result

After writing a program in Java to generate Fibonacci numbers using a recursive algorithm, I noticed the time increase in each iteration is approximately $\Phi$ times greater than the previous. ...
1
vote
2answers
86 views

Need help understanding Fibonacci Fast Doubling Proof

From this website, http://www.nayuki.io/page/fast-fibonacci-algorithms (fast doubling proof close to the bottom of the page). I have understood the proof for the most part but I am struggling to see ...
2
votes
0answers
66 views

Transforming the cubic Pell-type equation for the tribonacci numbers

The Lucas and Fibonacci numbers solve the Pell equation, $$L_n^2-5F_n^2=4(-1)^n\tag1$$ The tribonacci numbers $z = T_n$ are positive integer solutions to the cubic Pell-type equation, $$27 x^3 - 36 ...
9
votes
2answers
183 views

A number $N$ is a $k$-nacci number if and only if …

For $k\ge 2\in\mathbb N$, one can define the $n$-th $k$-nacci number $f_k(n)\ (n=0,1,\cdots)$ as $$f_k(0)=f_k(1)=\cdots=f_{k}(k-2)=0,\ \ ...
2
votes
1answer
29 views

How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about ...
2
votes
1answer
59 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 ...
0
votes
0answers
68 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
4
votes
1answer
58 views

Closed form as sum and combinatorial of Fibonacci numbers

How can I prove that the Fibonacci numbers that are defined as $F_n=F_{n-1}+F_{n-2}, \; n \geq 2$ and $F_0=0,\ F_1=1,\ F_2=1$ have the form: $$F_n=\sum_{k=0}^{n-1} \binom{n-1-k}{k}, \; n\ge 2 $$ I ...
3
votes
0answers
144 views

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$?

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$? By better comparison series than $\sum_0^\infty2^{-k}$ we mean a series $\sum c_k$ s.t. ...
1
vote
1answer
65 views

What is wrong with the following argument involving Fibonacci and Lucas numbers?

The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$. What is wrong with the following argument? Assuming $L_n = F_n$ for $n = ...
3
votes
4answers
152 views

A calculus proof for the general term of the Fibonacci sequence [duplicate]

Let $a_0=1$,$a_1=1$ and $a_n=a_{n-1} + a_{n-2}$ for $n \geq 2$, I would like to prove: $$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n + 1}- \left(\frac{1-\sqrt{5}}{2}\right)^{n + ...
3
votes
4answers
45 views

Prove $F^2_{n+1} - F_nF_{n+2} = (-1)^n$

This is a question about Fibonacci sequences, a sequence in which the previous terms build up upon the current term (e.g. $F_1 + F_2 = F_3$ where $F_1 = F_2 = 1$). How would I go about proving ...
0
votes
1answer
68 views

are there infinitely many primes in Fibonacci sequence

There is one proof about infinitude of prime with following method, http://www.ams.org/mathscinet-getitem?mr=2271540 Also it is well know that any two consecutive Fibonacci numbers are mutually ...
2
votes
4answers
129 views

Prove that $p$ divides $F_{p-1}+F_{p+1}-1$ [duplicate]

Given the Fibonacci sequence $(F_n)$, defined by $F_0=0,F_1=1, F_{n+2}=F_{n+1}+F_n$, and $p$ an odd prime number, how to prove that $p$ divides $F_{p-1}+F_{p+1}-1$? Is induction a good idea here? ...
2
votes
2answers
78 views

Sum of squares of Fibonacci Numbers

$$ \sum_{i=0}^{n} (F_{2i+1})^2 = \;?$$ I know that sum of squares of first $n$ Fibonacci numbers is $F_{n} \times F_{n+1}$.
0
votes
2answers
60 views

Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and ...
1
vote
1answer
38 views

Counting the sequences of coin flips that end HH after $n$ flips (a more efficient method?)

I figured out that for any given $n$ the number of sequences of heads and tails that satisfy the condition that HH wasn't flipped consecutively until flips $n-1$ and $n$ is equal to the $(n-1)$th ...