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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2answers
134 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 ...
2
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2answers
43 views

Proving that for Fibonacci numbers $a_n \lt (\frac {1 + \sqrt 5} 2)^n$ for $n \ge 1$

I'd like to prove that for Fibonacci numbers $a_n \lt \left(\frac {1 + \sqrt 5} 2\right)^n$ for $n \ge 1$. I suppose it needs induction so, after verifying the trivial case $n=1$, the inductive step ...
2
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4answers
46 views

Finding growth bounds on Fibonacci Sequence

I've been working on this following problem: Find a constant $c< 1$ such that $F_n \leq 2^{cn}$ for all $n \geq 0$. I honestly have no idea where to begin on this. I've done plenty of proofs ...
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2answers
323 views

Interpolated Fibonacci numbers - real or complex?

The common Binet-formula for the Fibonacci-numbers $$ f_n = {\varphi^n- (1-\varphi)^n \over \sqrt 5 } \small {\qquad \qquad \text{ where }\varphi={1+\sqrt 5\over 2}}$$ allows interpolation to ...
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0answers
87 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 ...
0
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1answer
39 views

Generalized Fibonacci Sequence with Modular Arithmetic

Consider the following generalized Fibonacci sequence: Given $a$ and $b$ positive integers, and the known values $g_1, g_2, ...g_b$ where $g_k = g_k$ (mod $a$), then for ...
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0answers
50 views

How to prove a claim about Fibonacci sequence

I have to prove that for any natural number $n$ there exists $i>0$ such that $n\mid F_i$, where $F_i$ is the $i$-th Fibonacci number.
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0answers
31 views

Generalized Fibonacci Sequences G_{n+p}

I have been given the following generalized fibonacci sequences: For some positive integer $m,p$, $g_{n+p}=g_{n+p-1}+g_{n+p-2}+...+g_{n+1}+g_n (mod m)$ I have been given two problems: (1) For $m=2$ ...
3
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2answers
54 views

Is fibonacci sequence a member of more broad family of sequences?

Yesterday, I was pondering on the Fibonacci sequence and I started to discover some features of it that were previously unknown to me. Such as, 1, 1, 2, 3, 5, 8, 13, 21, 34 .... 1 ) The nth element ...
2
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1answer
110 views

How do I choose first terms of a Fibonacci sequence?

Let $f(0)=a$ and $f(1)=b$ be the first two terms of a Fibonacci sequence. We know that this sequence is periodic in $\mod{p}$, where $p$ is a prime number, and the period of the sequence is $p-1$. I ...
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1answer
109 views

The totient of Fibonacci numbers is divisible by $4$

Let $\{f_i\}_{i\in\mathbb N}$ be the sequence of Fibonacci numbers, i.e. $1,2,3,5,8,13,21,34,55,\cdots$, For every integer $n\gt3$ prove that $$4\mid\phi(f_n)$$ where $\phi$ is Euler's totient ...
2
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1answer
48 views

Showing that this sum is equal to the fibonacci numbers

How do I show that the following sum is equal to the fibonacci numbers? Atleast numerical evaluation suggests it is $$ \sum_{k=0}^{\lceil n/2\rceil}\binom{n+1-k}{n+1-2k} $$ The image below shows how ...
3
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2answers
446 views

Fibonacci / Lucas Numbers Relationship: $F_{2n} = F_n L_n$

Prove the identity by induction: $$ F_{2n} = F_n L_n, $$ where $F_n$ and $L_n$ are the $n^{th}$ Fibonacci and Lucas number, respectively. I have an answer but am not happy with it since it doesn't ...
0
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1answer
34 views

Generalized Fibonacci Sequences with Modular Arithmetic

Consider the following generalized fibonacci sequence: For $m,p$ positive integers and $g_k =g_k (mod m)$, then for $n=1,2,3,...$ $g_{n+p}=g_{n+(p-1)}+g_{n+(p-2)}+...+g_{n+1}+g_n (modm)$ I need to ...
2
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1answer
69 views

Prove by induction that numbers of Fibonacci of the form $F_{3n}$ are even

I am not quite asking to solve this problem, but I am asking what they are asking me. This is the problem: The Fibonacci numbers $F_n$ for $n \in \mathbb{N}$ are defined by $F_0 = 0$, $F_1 = 1$, ...
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0answers
40 views

Problem about fibonacci sequence via quadratic roots in gelfand's algebra text.Need hints.

I have solved a preceding question proving that the common ratio of such a sequence is $ \frac {1+\sqrt{5}}{2} $ or $ \frac {1-\sqrt{5}}{2} $ (resolving a quadratic equation) . The present problem is ...
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0answers
59 views

Sum of Power of Two Fibonacci reciprocals

Evaluate $$\sum_{k=0}^{\infty} \frac{1}{F_{2^n}} \;.$$ I'm thinking of using a relation from a term to another.
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0answers
39 views

A binary plot of the Catalan numbers and the pseudo-Fibonacci series that can be found inside. Why do they appear?

I was trying to find in Internet a binary plot of the Catalan numbers, and I did not find anyone... so I did it by myself and here it is (about 2000 elements): There are not clear patterns inside ...
2
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2answers
67 views

Sum of Fibonacci numbers

While trying to find find a formula to calculate the length of the golden spiral I came across the sum of the Fibonacci numbers. I noticed that $$\text{Fibonacci numbers: }1,1,2,3,5,8,13,21,34...$$ ...
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3answers
65 views

Fibonacci induction proof?

The Fibonacci Numbers $(f_n)$ are defined $f_1=f_2=1$, and $f_n=f_{n-1}+f_{n-2} ,\,\,\,\forall n \geq2$. Prove that for every integer $n \geq 1$, $$f_1 +f_2 +···+f_n =f_{n+2}−1$$
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2answers
51 views

Prove the following recurrence: $F_{2n+1}=3F_{2n-1}-F_{2n-3}$

Prove the following identity: $$F_{2n+1}=3F_{2n-1}-F_{2n-3}$$ So far I know that $F_n=F_{n-1}-F_{n-2}\implies F_{2n+1}=F_{2n}+F_{2n-1}$ Just not sure where to go from here to get to the conclusion. ...
0
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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 ...
3
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4answers
817 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
1
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1answer
41 views

Hadamard's product of Fibonacci generating functions.

$F(s) = \frac{1}{1-s-s^2}=\sum_{n\geq0}F_ns^n$. I want to calculate $F(s) \circ F(s) = \sum_{n\geq0}F_{n}^2s^n$. I have tried using Binet"s formula, but problem remains unsolved.
2
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0answers
86 views

Proving $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $ for Fibonacci numbers

How can I prove the following identity about the Fibonacci numbers by using matrices or determinants? $ F_{m}F_{n}=\dfrac{1}{5}(L_{m+n}-(-1)^{n}L_{m-n}) $
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1answer
37 views

Fibonacci Number Formula for nth term [duplicate]

Hey is there any known combinatorial formula for nth fibonacci number? (n+1)th fibonacci number is given by summation of r=0 to (round)n/2:C(n-r,r) Can someone verify the formula?Help!
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1answer
747 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) ...
2
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1answer
38 views

Help with how to prepare the inductive step of a strong induction exercise.

I have the following exercise: "Use strong induction to prove that $f_1^2 + f_2^2 + \cdots + f_n^2 = (f_n)(f_{n+1})$ where $f_n$ in the nth Fibonacci number." This is what I have done: Fibonacci ...
0
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0answers
21 views

Can't find ANY golden ratio in the schroder house…

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to ...
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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
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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. ...
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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
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1answer
20 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 ...
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3answers
2k views

Fibonacci sequence divisible by 7?

Make and prove a conjecture about when the Fibonacci sequence, $F_n$, is divisible by $7$. I've realized it's when $n$ is a multiple of $8$. I just don't know how to go about proving it.
8
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8answers
525 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...
2
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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 = ...
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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
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1answer
90 views

Hyper sum of Fibonacci numbers

Let $F(n)$ be the $n$-th Fibonacci number. That is, $F(n)$ satisfies $F(0)=0,F(1)=1,F(n)=F(n−1)+F(n−2) (if n≥2).$ Let $f_k(n)$ be the function such that $f_0(n) = F(n)$, $f_k(n) = ...
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15answers
12k views

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 ...
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1answer
305 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 ...
0
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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
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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 $f(n) = f(n+2) - f(n+1)$ ...
0
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2answers
37 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 ...
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3answers
85 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 ...
2
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4answers
950 views

Strong inductive proof for this inequality using the Fibonacci sequence.

Problem I need to determine for what natural numbers is $2n < F_n$, where $F_n$ is the $n^{th}$ Fibonacci number determined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1}+F_{n-2}$. I then need to ...
3
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2answers
141 views

Fibonacci combinatorial identity

Can someone explain how to prove the following identity involving Fibonacci sequence $F_n$ $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$ ?
8
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2answers
818 views

Proof by induction on Fibonacci numbers: show that $f_n\mid f_{2n}$

I was studying Mathematical Induction when I came across the following problem: The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation- $f_n = f_{n-1} + ...
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
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0answers
159 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
2answers
163 views

Prove the $n$th Fibonacci number is the integer closest to $\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n$

Prove that the $n$th Fibonacci number $f_n$ is the integer that is closest to the number $$\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n.$$ Hi everyone, I don't really understand the ...