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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Remarkable relation between Fibonacci numbers and its squares!

There is a remarkable relation between Fibonacci numbers and its squares: $F^{2}_{n} +F^{2}_{n+1}=F_{2n+1}$. I know how to prove it using $F_{n}=\frac{\sqrt{5}}{5}((\frac{1+\sqrt{5}}{2})^n -(\...
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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
174 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
293 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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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: $...
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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
1k views

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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1answer
3k views

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 ...
9
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1answer
189 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 ...
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2answers
302 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 ...
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3answers
86 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 ...
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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 ...
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0answers
117 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: $$F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n-\...
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5answers
781 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: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
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2answers
125 views

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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1answer
127 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} ...
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0answers
73 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
81 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 ...
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1answer
112 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 ...
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2answers
672 views

Inductive proof of a formula for Fibonacci numbers

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: $$F_n=\frac{1}{\sqrt{5}}⋅\left(\frac{1+\sqrt{5}}{2}\right)^n-\...
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1answer
44 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 ...
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2answers
106 views

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 $F_n=F_{n-...
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1answer
94 views

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
77 views

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
70 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. $$\left(\frac{1+\sqrt{5}}{2}\...
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2answers
388 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 words,...
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2answers
299 views

An analogue of Hensel's lifting for Fibonacci numbers

Let $F_0, F_1, F_2, \ldots$ be the Fibonacci numbers, defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n\geq 2$. In this question Oleg567 conjectured the following interesting ...
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4answers
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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
318 views

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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0answers
112 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: Show with induction, that the $i^{\rm th}$ Fibonacci number satisfies the equality: $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
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1answer
58 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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3answers
421 views

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} F_{n+...
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1answer
135 views

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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2answers
321 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
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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 ...
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4answers
2k views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
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1answer
81 views

Fibonacci numbers $F_{n+3} + F_{n} = 2F_{n+2}$

Prove $F_{n+3} + F_{n} = 2F_{n+2}$ for any positive integer n. So What I did was this: fn+ fn+1 = fn+2 fn + fn+1 = fn+2 => fn+2 -fn+1 fn+1 + fn+2 = fn+3 then I subsituted into equation in ...
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4answers
179 views

Fibonacci Calculation using a larger matrix

So the formula to generate the fibonacci sequence in matrix form is: $$ \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{...
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5answers
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How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or $...
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1answer
1k views

Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
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2answers
82 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
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1answer
464 views

Asymptotic value of Fibonacci numbers

It is well known that $F_n\sim\frac{\phi^n}{\sqrt{5}}$, where $\phi=\frac{1+\sqrt{5}}{2}$. Does someone know a better estimate? With proof please. I'm trying to calculate the following limit: Let $...
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1answer
84 views

Let $u_n$ be the $n$-th entry in the Fibonacci sequence $1,1,2,3,5,8,13,\ldots$

If you start with $u_1 = 1$ and $u_2 = 1$, then the sequence can be generated using the formula $$u_{n+1} = u_n + u_{n-1}\ .$$ If $u_n = r^n$, what is r? Can anyone figure this out? I am so stuck ...
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2answers
721 views

The sum of $n$ consecutive Fibonacci numbers.

The sum of $8$ consecutive Fibonacci numbers is divisible by $3$. How can I generalize this for the sum of $n$ consecutive Fibonacci numbers? For example, $$1+1+2+3+5+8+13+21=54=3\times 18 \\ ...
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1answer
111 views

Proof that golden angle successively divides the largest gap by the golden ratio?

The golden angle divides the circumference of a circle by the golden ratio. "If radial spokes are placed successively into the circle, each spaced by a golden angle increment, then each additional ...
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1answer
96 views

What's the Lucas version of the Möbius test for Fibonacci numbers?

I recently came across the following, attributed to Möbius: $$(a\in\mathbb N)=F_n\iff\left[\varphi a-\tfrac{1}{a},\varphi a+\tfrac{1}{a}\right]\ni(b\in\mathbb N)$$ It is the lesser-known test used to ...
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2answers
110 views

How do I apply the $\pm4$ part of the equation $5F_n^2\pm~4=L_n^2$ without knowing $n$?

I'm trying to test a great many numbers $a^3+b^3$ to see if any of them are Fibonacci using the formula $$a^3+b^3=F_n \iff 5(a^3+b^3)^2\pm~4=L_n^2$$ I want to make my search more efficient by having ...
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1answer
62 views

Looking for $\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$

I am looking for the sum of the series: $$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}}$$ where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the ...
1
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1answer
169 views

What is length of period of fibonacci number mod 1000033

Can some explain me how the period of Fibonacci mod $1000033$ is $4684$. As we know if $n$ mod $5$ is $2$ or $3$ then period is $2n + 2$ so the period should me $2\times1000033 + 2$ but why it is $...
2
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1answer
51 views

finding $n$ consecutive composite Fibonacci numbers.

For each $n$, How can we find $n$ consecutive composite Fibonacci numbers?