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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how to calculate a modified fibonacci via matrix exponentiation

If I modify the fibonacci recurrence to be the following way: f(0) = 1 f(1) = 1 f(N) = f(N - 1) + f(N - 2) + 1 Is it possible to represent this recurrence in a matrix equation similar to the one ...
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0answers
70 views

geometric proof for fibonacci numbers identity with sum of two squares

Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares The link above gives the induction proof does a geometric proof using the squares with Fibonacci length exist for this?
2
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4answers
159 views

Find the sum of an infinite series of Fibonacci numbers divided by doubling numbers. [duplicate]

How would I find the sum of an infinite number of fractions, where there are Fibonacci numbers as the numerators (increasing by one term each time) and numbers (starting at one) which double each time ...
4
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0answers
30 views

Symmetric Fibonacci images

I was playing with the Turtle module in Python and decided to try plotting the Fibonacci series with the following scheme where $f_n$ is the $n^{th}$ Fibonacci number: Rotate the turtle $k (f_n \...
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0answers
56 views

Proof of a formula for generalized Fibonacci numbers

I have done the verification for $$U_rU_{n−1} − U_{r−1}U_n = (−1)^{r−1}U_{n−r}$$ I realized when I was doing for $n=k+1$, the expression $U_rU_k − U_{r−1}U_{k+1}$ would not equate to $(−1)^{r−1}U_{n−r}...
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1answer
47 views

How to find $\sum_{n = 0}^{ \infty} \frac{F_n}{3^n}$

How can I find $$\sum_{n = 0}^{ \infty} \frac{F_n}{3^n}$$ If I know that the generating function for the Fibonacci sequence is $G(t) = \frac{t}{1 - t - t^2}$?
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2answers
88 views

Proof a number is Fibonacci number

I have a question regarding the proof that a number n is a Fibonacci number if and only if $5n^2-4$ or $5n^2+4$ is a perfect square. I don't understand the second part of the proof: knowing that $5n^2-...
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4answers
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Proof of ${F(n+4)}^{4} - {4F(n+3)}^{4} - {19F(n+2)}^{4} - {4F(n+1)}^{4}+{F(n)}^{4} = -6$

Observe: \begin{matrix} F(n)|&{F(n)}^{4}& - {4F(n+1)}^{4}& - {19F(n+2)}^{4}&- {4F(n+3)}^{4}&{F(n+4)}^{4}& = -6\\ 1|& 1& -4& -304& -324& 625&=-6\\ 1|&...
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0answers
526 views

proving Fibonacci numbers using mathematical Induction?

Can anyone confirm whether my answer is correct, please. Let suppose we have the following fibonacci numbers as shown: $f(0) = 0, f(1) = 1$, and $f(n) = f(n-1) + f(n-2)$ for $n \geq 2$. Prove that ...
0
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1answer
129 views

Prove that for each Fibonacci number $f_{4n}$ is a multiple of $3$. [duplicate]

The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \ge 2$. Prove that for each $n \ge 0$, $f_{4n}$ is a multiple of $3$. I've tried to prove to ...
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6answers
1k views

A Non-recursive Fibonacci Sequence

How can I determine the general term of a Fibonacci Sequence? Is it possible for any one to calculate F2013 and large numbers like this? Is there a general formula for the nth term of the Fibonacci ...
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2answers
71 views

Determine whether $A215928(n)=G_n$.

Let $G_0=1$ and $G_{n+1}=F_0G_n+F_1G_{n-1}+\cdots+F_nG_0$, where $F_n$ is the $n$th term of the Fibonacci sequence, i.e., $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. Let $P_0=P_1=1,\ P_2=2,$ and $P_{n+1}=...
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5answers
56 views

Prove by Induction $F_{2n} = F_{n} * L_{n}$, for n >= 1

Where $F$ is the Fibonacci Sequence, and $L$ is the Lucas Sequence. I need to find the inductive proof of this statement. I've got nearly a page of work in front of me trying to use definitions such ...
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2answers
113 views

Mathematical induction used on Fibonacci Sequence

I have no clue how to go about doing this question using induction. It states that the Fibonacci sequence is defined as: F0 = 0 F1 = 1 Fn = Fn-2 + Fn-1 for n>=2 Let S(n) = Fo + F1 + F2 +...+ Fn....
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3answers
157 views

Fibonacci sequence: how does $0$ get to $1$?

In the Fibonacci sequence, how does $0$ get to $1$? $$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \ldots$$ The rule is adding the previous $2$ numbers, and the previous $2$ numbers before $1$ are $0$ and $-1$...
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0answers
129 views

Arctangents, Fibonacci numbers, and the golden ratio

In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice ...
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3answers
60 views

What sequences where the difference between their consecutive terms is always a fibonacci numbers?

What sequence where the difference between its consecutive terms is always a fibonacci numbers ? I am trying to figure out a pattern in this sequence : 1,2,4,7,12,20,33,54,88
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1answer
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How many subsets does the set $\{1, 2, \dots , n\}$ have that contain no two consecutive integers if $1$ and $n$ also count as consecutive?

How many subsets does the set $\{1, 2, \dots , n\}$ have that contain no two consecutive integers if 1 and n also count as consecutive? It looks that the number of such subsets obeys the (...
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2answers
247 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 \...
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1answer
684 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
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2answers
83 views

Trouble with Fibonacci number mathematical induction

The problem is: $$F_n \leqslant 2F_{n-1}\quad\text{for every integer} \quad n \geqslant 2.$$ I got the smallest case, I just don't know how to get the assumption and the rest of it
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0answers
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Legitimate papers refuting the significance of the golden ratio in art?

I'm not sure this is the right place to ask about this, but is there any legitimate peer-reviewed paper refuting the significance of the golden ratio in art? I can find numerous websites and blogs ...
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2answers
65 views

Number of Fibonacci series that contain a certain integer

In my question, I consider general Fibonacci sequences (sequences satisfying the recurrence relation $F_{n+2}=F_{n+1}+F_n$ independent of their starting value). Given two arbitrary different integers, ...
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2answers
805 views

Fibonacci sequence - how to prove $a_n=\frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$ without induction [closed]

How to prove that $$a_n=\frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$$ without using induction?
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4answers
215 views

How to find closed form of summation of Fibonacci Sequence?

I created two formulas to prove a binary theory involving the Fibonacci sequence. (1) $\sum_{i=0}^n F_{2i+1} $ Equation (1) is the sum of all Fibonacci numbers up to $F_n$ where every $i$ in $F_i$ ...
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1answer
68 views

Fibonacci numbers identity - proof by induction

$\displaystyle F_{k-1} F_{k+1} - F_k^2 = (-1)^k$ I have done the base step for $k=1$ and it works. I realize we need to prove for $k+1$, so: $$F_k F_{k+2} - F_{k+1}^2 = (-1)^{k+1}$$ Could ...
2
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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 ...
0
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2answers
118 views

number theory fibonacci

Using facts of the Fibonacci sequence, I need to show that if $m,n$ are natural numbers that satisfy $m \mid F_n$ and $m \mid F_{n+1}$, then $m=1$. I am not sure where to start with this.. I am ...
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3answers
308 views

Fibonacci divisibility

Is $2051$ a factor of any fibonacci number? It is not a factor of any perfect number. The prime factors of $2051$ are $7$ and $293$, which are both prime. the $8$th fibonacci number, is the first ...
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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 ...
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1answer
79 views

Write out the Leutonian numbers that represent the first 12 positive integers.

How could I write out the leutonian numbers that represent the first 12 positive integers ? I have no idea how to start.
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1answer
106 views

Fibonacci numeration system

Instead of binary or decimal, the Kingdom of Leutonia uses an unusual system to represent numbers, based on the Fibonacci sequence. The Fibonacci sequence $F_0,F_1,F_2,\dots$ is defined recursively as ...
3
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1answer
130 views

Powers of 2 in the product of the Fibonacci numbers

I'm working on finding a summation that pulls the powers of 2 out of the product of the Fibonacci numbers. I've noticed some patterns for the Fibonacci number. For example. Looking at the Fibonacci ...
2
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1answer
114 views

determine the number of terms in a fibonacci sequence that are divisible by $3$

Consider the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, . . .$ where each term, after the first two, is the sum of the two previous terms. How many of the first $1000$ terms are divisible by 3?
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1answer
220 views

Are the Fibonacci numbers' prevalence in nature due to confirmation bias?

The Fibonacci numbers are frequently found in nature, such as in the petals of flowers or the shape of pinecones. But are the numbers actually any more prevalent than other numbers? Could it all be ...
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1answer
502 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
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1answer
85 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
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0answers
297 views

Prime number distribution theory for dummies

For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis http://en.wikipedia.org/wiki/Riemann_hypothesis#...
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0answers
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Fibonacci sequence: Prove the formula $f_{2n+1}=f_{n+1}^2 + f_n^2$ [duplicate]

I can't seem to figure out this proof. I'm using weak induction and always get stuck during the inductive step. Prove for n > 0: $$f_{2n+1} = f_{n+1}^2 + f_n^2$$
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3answers
175 views

When do you need to prove more than 1 base case in the Fibonacci problem?

I was trying to prove that if $F_n = F_{n-1} + F_{n-2}$ and $F_1 = 1$ and $F_2 = 1$, then the following proposition $P(n)$ was true $\forall n : \sum^n_{i=1}F_i=F_{n+2} - 1$ The issue I have with the ...
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0answers
34 views

How many rationals for a given $n \in \Bbb N \;\backslash \{1\}$?

Fix $n \in \Bbb N, n> 1$. Now choose a two digit base-$n$ number $ab $ say. There's $n^2$ choices for this. Consider the number $0.c_1 c_2 c_3 \ldots$ where the $c_i$ are defined recursively: $...
33
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4answers
1k views

Conjecture: Only one Fibonacci number 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 $...
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1answer
50 views

Number of strings of size $k$ that do not have 'ab'

Consider $\Sigma = \{a,b,c\}$ and the language $L$, the set of all strings that do not contain 'ab' Find strings, of size $k$ is in $L$ ($L_k$) Consider $A_k$ (strings of size $k$ that end in $a$) ...
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1answer
193 views

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 ...
6
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3answers
88 views

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 ...
2
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4answers
271 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 ...
72
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4answers
5k 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 ...
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1answer
43 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 ...
8
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2answers
3k 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 5}{...