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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2
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2answers
160 views

Sum $\frac{1}{1\times2}+\frac{1}{1\times3}+\frac{1}{2\times5}+\frac{1}{3\times8}+\cdots$

If $f_n$ is the Fibonacci series, with $1,1,2,3,5,8,\ldots$ prove that $$\sum_{i=2}^\infty\frac{1}{f_{i-1}\cdot f_{i+1}} = 1$$ So my idea was to try to convert this series into a telescoping sum ...
6
votes
4answers
841 views

Fibonacci numbers and golden ratio: $\Phi = \lim \sqrt[n]{F_n}$

Let $\Phi$ be the golden ratio and $F_n$ be the usual Fibonacci numbers. How can I derive the following formula? $$ \Phi = \lim_{n\rightarrow \infty} \sqrt[n]{F_n} $$ I know the usual relation $$ \...
1
vote
3answers
107 views

Prove the given property of the Fibonacci numbers

I found in one of the books I read a lot of interesting properties of fibonacci numbers and among others this one in particular: For all $n \in \mathbb N$, $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. I ...
1
vote
2answers
176 views

Limit of ratio of successive n-nacci numbers?

The n-nacci numbers are defined as $${}_nF_k = {}_nF_{k - 1} + {}_nF_{k - 2} + \cdots + {}_nF_{k - n + 1}$$ Now, it's pretty well-known that the limit of successive $2$-nacci numbers (i.e. the ...
2
votes
2answers
145 views

What is the relation between this binary number with no two 1 side by side and fibonacci sequence?

I saw this pattern of binary numbers with constraints first number should be 1 , and two 1's cannot be side by side. Now as an example ...
1
vote
2answers
361 views

Generating function for squared fibonacci numbers

We know that generating function for fibonacci numbers is $$B(x)=\frac{x}{1-x-x^2}$$ How can we calculate $B(x)^2$? I thought that, if we have $B(x)=F_n*x^n$ then $$B(x)*B(x) = \sum_{n=0}^\infty (\...
1
vote
1answer
64 views

Generating function for kind of sum of Fibonacci numbers

Let's have a sequence $$a_n = \sum_{i=0}^n F_iF_{n-i}$$ where $F_n$ is n-th Fibonacci number. I tried to solve it somehow, but i'm pretty stuck. Defining Fibonacci numbers $$b_0=0, b_1=1, b_n=b_{n-1}...
1
vote
1answer
50 views

How do I calculate the number of members in a limited Fibonacci series? [duplicate]

Looking for an algorithm that will give me the number of members that will result from calculating a Fibonacci series, given a particular limit. For example, if I start the series at 1 and limit my ...
0
votes
1answer
83 views

Generalisation of Fibonacci

Somehow a generalisation of the fibonacci numbers, do numbers created by the formula $ F(n) = F(n-1) + [F(n-1)-F(n-2)+F(n-3)-F(n-4)+F(n-5)-F(n-6).....]$ with $F(1) = 1$ have a specific name?
0
votes
2answers
88 views

Fibonacci Sequence and odd/even addition [duplicate]

Prove that f0 – f1 + f2 - … - f2n-1 + f2n = f2n-1 – 1. For n is all positive numbers. I have an idea to what I must do, but I can't figure what the base case is. I think it is f(0) = 0 and f(1) = 1. ...
2
votes
1answer
133 views

No advantage to the closed form for Fibonacci numbers?

The closed forms for the Fibonacci sequence, such as: $$F_n=\frac{\varphi^n-\widehat\varphi^n}{\sqrt5}=\frac{\varphi^n}{\sqrt5}-\frac{\widehat\varphi^n}{\sqrt5}\;,$$ the Binet formula, do not seem ...
2
votes
1answer
153 views

Question on Proof that the Fibonacci Word is Sturmian

I am currently reading a text where it is proved that the infinite Fibonacci Word $u$ defined as the limit of the sequence $$ u_n = \varphi^n(0) $$ where the morphism is given by $\varphi(0) = 01, \...
6
votes
4answers
117 views

Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers

Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$. For example: $32 = 21 + 8+3 = F_8+F_6+F_4$ Research ...
2
votes
2answers
93 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}) / ...
-1
votes
2answers
519 views

how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
0
votes
1answer
681 views

Fibonacci proof by Strong Induction

Prove by strong induction that for a ∈ A we have $F_a + 2F_{a+1} = F_{a+4} − F_{a+2}.$ $F_a$ is the $a$'th element in the Fibonacci sequence
1
vote
5answers
621 views

Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$ [duplicate]

We define the $n^{th}$ Fibonacci number as $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. Consider $$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}. $$ I wrote a script and found that this ...
49
votes
5answers
5k views

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 ...
0
votes
2answers
183 views

Prove that a Fibonacci number is greater than $ φ^n$

How can I prove the following: If $f_n$ is a number of the Fibonacci sequence and φ= $\frac{1+\sqrt{5}}2$, then $f_n > φ^n$ for every $n >2$? I have tried using induction but I can't ...
1
vote
3answers
417 views

partial Fibonacci summation

Let $F_{n}$ be the n-th Fibonacci number. How to calculate the summation like following: $\sum_{n \geq 0} F_{3n} \cdot 2^{-3n}$
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2answers
92 views

Proof by mathematical induction - Fibonacci numbers and matrices

Using mathematical induction I am to prove: $ \left( \begin{array}{ccc} 1 & 1 \\ 1 & 0 \end{array} \right)^n $ = $ \left( \begin{array}{ccc} F_{n+1} & F_n \\ F_n & F_{n-1} \end{...
4
votes
2answers
1k views

Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate]

Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists. How can we prove ...
1
vote
1answer
309 views

Calculate Number of ways to make the grid

We wish to tile a grid of size Nx2 with rectangles (dominoes) of 2x1 (in either orientation).For given N I need to find the number of different ways to tile the grid. EXAMPLE : For N=1 answer is 1 ...
0
votes
0answers
60 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
2
votes
3answers
306 views

How to go about solving this recurrence relation?

In my discrete math class we were given the problem of finding an explicit formula for this recurrence relation and proving its correctness via induction. $$a_n=2a_{n-1}+a_{n-2}+1, a_1 = 1, a_2=1.$$ I ...
2
votes
4answers
266 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 ...
5
votes
5answers
2k views

Where do the first two numbers of Fibonacci Sequence come from? [duplicate]

I'm trying to code a simple algorithm that prints out the $n^{th}$ Fibonacci number. However, my program requires the initial seed values $F_0 = 0$ and $F_1 = 1$, when I'm hopeful I can figure ...
0
votes
1answer
49 views

asymptotics of $F_n = F_{k-1} F_{n-k-1} + F_{k-1} F_{n-k} + F_{k-2} F_{n-k} $

Starting from $F_n = F_{k-1} F_{n-k-1} + F_{k-1} F_{n-k} + F_{k-2} F_{n-k} $ and letting $F_k \approx \phi^k$, I am hoping to find the corresponding statement for the Golden ratio: $\phi^n = 2 \...
2
votes
2answers
167 views

Prove that Binet's formula gives an integer, using the binomial theorem

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$. The textbook states that it's equal to the $n$-th Fibonacci ...
6
votes
1answer
693 views

How to prove that Fibonacci number is integer?

How to prove that formula for Fibonacci numbers are always integers, for all $n$: $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{...
1
vote
1answer
657 views

Proof by induction involving fibonacci numbers

The Fibonacci numbers are defined as follows: $f_0=0$, $f_1=1$, and $f_n=f_{n−1}+f_{n−2}$ for $n≥2$. Prove each of the following three claims: (i) For each $n≥0$, $f_{3n}$ is even. (ii) For each $n≥...
2
votes
0answers
104 views

Inequality with Fibonacci numbers

The sequence $F_n$ of natural numbers defined by equation $F_{n+2}=F_{n+1}+F_{n}$, with $F_0=0, F_1=1$ is called the Fibonacci sequence. The n-th term in the sequence is called the n-th Fibonacci ...
5
votes
2answers
224 views

Number of bitstrings with $000$ as substring

I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$? Now there are many ways to go about this but if I choose starting ...
5
votes
1answer
567 views

Sum of digits in Fibonacci sequence

This is my first question here so please go easy on me. If you add the digits of each number on the Fibonacci sequence until your number is less than 10, it seems that you get a pattern of 24 numbers ...
5
votes
4answers
2k views

How to prove this is a rational number

I'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me? $$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$
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2answers
1k views

Formula for binary sequences of length m with no n consecutive 1s?

Formula for binary sequences of length $m$ with no $n$ consecutive $1$s? I know The number of binary strings of length $m$ without consecutive $1$s is the Fibonacci number $F_{m+2}$. But how about ...
8
votes
1answer
180 views

Is this number rational or irrational?

Start writing down the Fibonacci numbers, using two digits for each one 01 01 02 03 05 08 13 21 34 55 ... Eventually you will reach three digit numbers. When ...
4
votes
1answer
271 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
26
votes
3answers
1k 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 ...
0
votes
2answers
454 views

Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
4
votes
1answer
122 views

Fibonacci numbers expressed as squares of lower Fibonacci numbers

I am no mathematician so my apologies for my ignorance. I notice that every number in the Fibonacci series can be expressed as a previous Fibonacci number squared plus or minus (alternating) another ...
2
votes
1answer
86 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: ...
2
votes
2answers
121 views

Internet problem solving contest question

I am trying to solve a problem from the IPSC http://ipsc.ksp.sk/2001/real/problems/f.html It basically asks to compute the following recursion. ...
0
votes
2answers
49 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 = ...
1
vote
1answer
57 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
votes
0answers
120 views

Lucas Number Equivalent of the Pisano Period?

I'm doing some work with the Pisano Period, and it's leading to also talk about the Lucas numbers $mod \ m$, and their period. Is there a specific notation already designated? Sticking with using a ...
2
votes
4answers
113 views

New Fibonacci sequence

I have new Fibonacci number That I want to know is there any special direct formula to count f(n). like the normal Fibonacci: F(0) = 7, F(1) = 11, F(n) = F(n-1) + F(n-2) (n>=2) For example I want to ...
3
votes
1answer
67 views

All pairs sum to a different value

If we consider the integers $\{1,\dots,n\}$, what is the size of the largest subset $A$ so that all distinct pairs $x, y \in A$ sum to a different value? For this to make sense $(x,y)$ and $(y,x)$ ...
9
votes
1answer
1k views

Proof that Fibonacci Sequence modulo m is periodic? [duplicate]

It's well known that the Fibonacci sequence $\pmod m$ (where $m \in \mathbb N$) is periodic. I have figured out a proof for this, but upon googling, I found proofs online that were far more ...
4
votes
3answers
353 views

Probability that a chosen number will be a Fibonacci number

Suppose that I randomly choose an integer $x$ with $1 \leq x \leq n$ where $n$ is a natural number. What is the probability that $x$ will be a Fibonacci number?