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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1answer
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Relationship between Primes and Fibonacci Sequence

I recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connection ...
7
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1answer
62 views
+200

If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
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16answers
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Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
0
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2answers
41 views

Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
7
votes
1answer
315 views

Construction of generating function from identity

I am trying to solve identity involving binomials and fibbonaci numbers by using generating functions: $$\sum_{k=0}^n{n \choose k}{n+k\choose k}f_{k+1}=\sum_{k=0}^n{n \choose k}{n+k\choose ...
4
votes
3answers
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Fibonacci sequence: Given $n$ and $\mathrm{Fib}(n)$, is it possible to calculate $\mathrm{Fib}(n-1)$?

Given $n$ and $\newcommand{\Fib}{\mathrm{Fib}} \Fib(n)$, is it possible to calculate the previous number in the Fibonacci sequence - $\Fib(n-1)$ using integer math in constant time? In other words, I ...
33
votes
4answers
800 views

Conjecture: There's only one Fibonacci number that 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 ...
2
votes
4answers
120 views

For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$

For the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_n+F_{n+1}$, prove that $$\sum_{i=1}^n F_i= F_{n+2} - 1$$ for $n\ge 1$. I don't quite know how to approach this problem. Can someone help and ...
45
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4answers
3k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
1
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1answer
31 views

How do I proceed from here on finding the Binet's formula via generating functions?

So, I'm stuck with the algebra for the nth number on the Fibonacci sequence in here. I managed to get to the part where $G(x) = \frac{x}{1-x-x^2}$ $=$ $\frac{x}{(1-\alpha x)(1-\beta x)}$, and I know ...
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0answers
107 views

Solve fibonacci equation

Given two numbers M and N we need to solve : ( ∑ ( 6 * x * y * z * Fibo[x] * Fibo[y] * Fibo[z] ) ) % M , where x + y + z = N. Here x, y, z ≥ 0 and are integers. ...
2
votes
1answer
220 views

Cycles in the Generalized Fibonacci Sequence modulo a Prime

Suppose I have a fibonacci sequence 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 Now if I have a modulo 5 fibonacci sequence,it will look like ...
-1
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1answer
56 views

using induction to prove that the formula for finding the n-th term of the Fibonacci sequence is: [duplicate]

May someone help me? I am trying to use induction to prove that the formula for finding the $n^{th}$ term of the Fibonacci sequence is: ...
3
votes
2answers
595 views

Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt ...
3
votes
3answers
76 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
1
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2answers
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nth convolved Fibonacci numbers of order 6 modulo m

Problem: Find the coefficient of xk in (1−x−x2)-6 modulo m. Constraints: k≤264 m≤105, m can be a composite number. I have 10^5 such queries to process in 2 sec, so O(log k) for each query ...
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5answers
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Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
2
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1answer
97 views

Generalised Fibonacci

How to find the nth term of the recurrence in $\log n$ time. $$ \begin{array}{rcl} F[n]&=&F[n-1]+F[n-3]\\ F[2]&=&1\\ F[3]&=&2\\ F[4]&=&3\\ F[5]&=&4 \end{array} ...
11
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1answer
341 views

Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?

$\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$ (Link), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ \ldots )$ This is ...
11
votes
4answers
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The generating function for the Fibonacci numbers

Prove that $$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$ The coefficients are Fibonacci numbers $\left\{1,1,3,5,8,13,21,...\right\}$.
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3answers
142 views

A sum involving Fibonacci numbers, $\sum_{k=1}^\infty F_k/k!$

Let $F_k$ be Fibonacci numbers. I am looking for a closed form of the sum $\sum_{k=1}^\infty F_k/k!$. I tried to use Wolfram Alpha, but it is not doing the sum Fibonacci[k]/k! , k=1 to infinity. ...
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0answers
68 views

Sum of a Sequnce

How to find this sum $$ \sum_{i+j+k=n} \ G_i \cdot G_j \cdot G_k \ for \ all \ i,j,k > 0, $$ $$ G_i = i \cdot F_i, $$ where Fi - ith number Fibonacci, F0=0, F1=1
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3answers
71 views

Adding Fibonacci Numbers

I am getting confused on adding Fibonachi numbers. For example I know that: $\mathrm{F}_\mathrm{K+1}+\mathrm{F}_\mathrm{K}=\mathrm{F}_\mathrm{K+2}$ But I believe my logic is flawed. The way i am ...
13
votes
3answers
654 views

Prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$

We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact, $$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
1
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1answer
76 views

General formula of Fibonacci look alike series

I'm trying to discover the general formula of a series defined with recursion: $$ a_1 = 2, a_2 = 3, a_3 = 4 $$ and $$ a_n = a_{n-1} + a_{n-3} $$ It looks like Fibonacci, but the starting points are ...
3
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2answers
139 views

Another Fibonacci identity

Here's a problem that is leading me in circles. Consider the Fibonacci number $F_n$ defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 2$. Prove that $F_{2n-1} = F_{n}^2 ...
4
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1answer
122 views

Fibonacci Sequence and series limits

1) Let $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$. How do you prove that $$\sum_{n=2}^\infty \frac1{F_{n-1} F_{n+1}} = 1$$ $$\sum_{n=2}^\infty \frac{F_n}{F_{n-1} F_{n+1}} ...
5
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4answers
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How to show that this binomial sum satisfies the Fibonacci relation?

The binomial sum $$s_n=\binom{n+1}{0}+\binom{n}{1}+\binom{n-1}{2}+\cdots$$ satisfies the Fibonacci relation. I failed to prove that $\binom{n-k+1}{k}=\binom{n-k}{k}+\binom{n-k-1}{k}$... Any ...
5
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2answers
227 views

Two sums with Fibonacci numbers

Find closed form formula for sum: $\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n} \frac{F_{2k}F_{n-k}}{10^n}$ Find closed form formula for sum: $\displaystyle\sum_{k=0}^{n}\frac{F_k}{2^k}$ ...
5
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2answers
227 views

Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) ...
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3answers
137 views

“Fat” sets of integers and Fibonacci numbers

Let us call a set of integers "fat" if each of its elements is at least as large as its cardinality. For example, the set $\{10,4,5\}$ is fat, $\{1,562,13,2\}$ is not. Define $f(n)$ to count the ...
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3answers
615 views

Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
2
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1answer
372 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
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2answers
216 views

find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
3
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2answers
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Fibonaaci Recurrence

This is an interesting question where we are trying to solve another recursion which has same tree structure as the given recursion and also has term similarities Given Data in question ...
0
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1answer
34 views

Initial values appear from nothing

This answer says that any casual sequence of the kind $y_n = y_{n-1} + y_{n-2} + y_{n-3} + \ldots $ will stay constant-0 because $y_0$ is a sum of zeroes, so is $y_1$ and the rest of the sequence. I ...
0
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1answer
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Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
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1answer
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Generalized Fibonacci Sequence

I'm having trouble with a problem I encountered while studying Number Theory. This problem comes from the book Number Theory by George E. Andrews. It defines a generalized Fibonacci sequence $F_1$, ...
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2answers
113 views

Fibonacci number ending with given sequence of digits

Related to this question: For any given sequence of digits, does a Fibonacci number exist ending with such sequence? If not, it would be nice to find the smallest counterexample. (in other ...
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2answers
41 views

Proving Fibonacci inequality

I didn't see a question regarding this particular inequality, but I think that I have shown by induction that, for $n>1$. I am hoping someone can verify this proof. ...
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3answers
372 views

Significance of starting the Fibonacci sequence with 0, 1…

DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic. QUESTION: What is the significance of ...
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1answer
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Order of convergence for the method of false position

I'm reading about the order of convergence of the method of false position and there is one tricky point in the proof I don't understand. The method itself for finding the minimum $x^*$ of a function ...
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1answer
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An analogue of Hensel's lifting for Fibonacci numbers

In this question Oleg567 conjectured the following interesting fact about Fibonacci numbers: $$ k\mid F_n\quad\Longrightarrow\quad k^d\mid F_{k^{d-1}n} $$ that can be regarder as an analogue of the ...
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2answers
125 views

Fibonacci series, which is most pure mathematically? [closed]

There are various methods of generating the Fibonacci series. I'm going to list 3 of them in this question. Method 1: f[n] = f[n - 1] + f[n - 2] Method 2: (more ...
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1answer
54 views

Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
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0answers
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Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
2
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1answer
62 views

Source for relationship between $d$-ary Fibonacci numbers and generalized golden ratio?

I am not a mathematician (but a computer scientist) and stumbled across the following in the analysis of an algorithm (Berthold Vöcking: How Asymmetry Helps Load Balancing). The author gives Knuth: ...
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3answers
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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
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1answer
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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 $n$ ...
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1answer
62 views

Checking a formula works for many numbers [closed]

Hello StackExchange users. I have discovered a formula which works out any Fibonacci number using the formula to work out the value of any cell in Pascal's triangle. It uses the sigma notation which I ...