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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456 views

The Principle of Mathematical Induction

The question is Let $( F_0, F_1, F_2,... )$ be the Fibonacci sequence defined by $F_0=0,\, F_1=1, and F_{n+1}=F_n+F_{n-1}$, n greater than or equal to 1. Prove the following identities. ...
0
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1answer
106 views

The relation between piano 12-scale and Fibonacci?

One of my books says there is a relation between the chromatic musical scale [CC#DD#EFF#GG#AA#BC] and the Fibonacci sequence. So...what's the relation?
3
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1answer
152 views

Monotonicity of the sequence $ ( F_n^{\frac{1}{n}} ) $, where $ ( F_n ) $ is the Fibonacci sequence

Let $ F_n = F_{n-1} + F_{n-2} $ with $ F_0 = 1 $, $ F_1 = 1 $ (the Fibonacci sequence). I would like to know whether $ F_n^{\frac{1}{n}} $ is monotonically increasing in $ n $. It is not difficult to ...
2
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0answers
111 views

Why is the reciprocal of the second Fibonacci number negative?

The second Fibonacci number is 1, so it's reciprocal should be 1, right? Why is it that I get $-1$ when I plug in $2$ for n in the reciprocal of Binet's equation ...
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2answers
77 views

What is the approx value of $f(50001)/f(50000)$ where $f(i)$ gives the value of the $i$-th number in the Fibonacci series?

What is the approx value of $f(50001)/f(50000)$ where $f(i)$ gives the value of the $i$-th number in the Fibonacci series?
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1answer
1k views

Smallest Fibonacci number having a common factor with a given number

We have a number $k$ and we have to find the smallest Fibonacci number that has common factor with it(except $1$). We also have $2 \leq k \leq 1,000,000$. The required Fibonacci number is guaranteed ...
8
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1answer
119 views

Prove that $\forall p \in \Bbb P;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$

Prove that $\forall p \in \Bbb P,n \in \Bbb Z^+;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$ and $F_{5^n} \equiv 0 \mod 5^n$, where $\left(\dfrac{5}p\right)$ is the Legendre ...
3
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2answers
248 views

Prove that if $F_n$ is highly abundant, then so is $n$.

Define $F_n$ to be the $n$th Fibonacci number, define $\sigma(n)$ to be the sum of the divisors of $n$, and call $n$ highly abundant if and only if $\sigma(n)>\sigma(m) \hspace{3mm} \forall ...
28
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2answers
1k views

Why do some Fibonacci numbers appear in an approximation for $e^{\pi\sqrt{163}}$?

It is rather well-known that, $e^{\pi\sqrt{43}} \approx 960^3 + 743.999\ldots$ $e^{\pi\sqrt{67}} \approx 5280^3 + 743.99999\ldots$ $e^{\pi\sqrt{163}} \approx 640320^3 + 743.999999999999\ldots$ Not ...
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5answers
659 views

Showing that $f_{2n+1}=f_{n+1}^2+f_n^2$.

I am trying to solve the following exercise: Let $f_1=1$, $f_2=1$, $f_{n+1}=f_n+f_{n-1}$, where $n\in\mathbb{N}$. Show that $f_{2n+1}=f_{n+1}^2+f_n^2$. I have not had much progress, but this is ...
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4answers
487 views

Divisibility and the Fibonacci sequence

While studying the Fibonacci sequence I encountered this problem in the handout, and I can not understand how to do it. Show that if the Fibonacci sequence has a term divisible by a natural number ...
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1answer
110 views

Prove that if $\sigma(F_a)<\sigma(F_b)$, where $1 \leq b < a$, then $\sigma(a)<\sigma(b)$ also

Prove that if $\sigma(F_a)<\sigma(F_b)$, where $1 \leq b < a$, then $\sigma(a)<\sigma(b)$ also. Here $\sigma(x)$ denotes the sum of the divisors of $x$ and $F_x$ is the $x$th Fibonacci ...
1
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1answer
85 views

About the percentage of the mutiples of a prime $p$ in Fibonacci sequence

Suppose that a sequence $\{f_n\}$ is defined as $$f_1=f_2=1, f_{n+2}=f_{n+1}+f_n\ \ (n\ge1).$$ Supposing that for a prime number $p$ and a natural number $N$,$$F_p(N)=\{\ n\ |\ n \in\mathbb N,\ n\le ...
5
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1answer
188 views

Prove $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$

Prove the identity: $F_{1}^{2}+F_{2}^{2}+\dots+F_{n}^{2}=F_{n}F_{n+1}$, where $F_i$ denotes a Fibonacci number. How can I prove it using a geometric approach?
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1answer
30 views

Prove that $F_{\sum_{i=1}^ka_i}\geq \prod_{i=1}^kF_{a_i}\forall a_i,k \geq 1$

Is there an elegant way to do this? I don't think it's particularly difficult, since $F_n \sim \frac{\phi^n}{\sqrt{5}}$, so we expect that $F_{\sum_{i=1}^ka_i} \sim ...
5
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1answer
174 views

Prove that $F_{n}^{2}+F_{n+1}^{2}=F_{2n+1}$ [duplicate]

Prove that $F_{n}^{2}+F_{n+1}^{2}=F_{2n+1}$ This identity holds for $n>=1$ Instead of using induction, how do I prove it in a geometry approach?
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1answer
71 views

Prove that $F_{x^{n+1}} \sim 5^{\frac{x-1}{2}}F_{x^n}^x \forall x,n \geq 1$, holding either variable constant while the other goes to infinity

I noticed from looking at the prime factorizations of some Fibonacci numbers that all those with an index equal to a power of 5 divided that power of five, a property not guaranteed by the strong ...
3
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1answer
79 views

A Generalized Fibonacci sequence

I have the following recurrence $$u_{n+1}=u_{n}+a^{N-n}u_{n-1}\quad n\ge 1$$ where $$N\ge 1,\quad u_1=u_0=1,\quad 0< a\le 1$$ I want to find out $u_N$. My Try: For $a=1$ this is just the Fibonacci ...
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2answers
129 views

Is Fibonacci sequence the minimum of unique pairwise sum sequence?

Let $(a_n)_{n=1}^\infty$ be a strictly increasing (condition added per earlier answer of Amitesh Datta) sequence of natural numbers where all pairwise element sums are unique. Can anyone prove or ...
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0answers
506 views

A conjecture about Lucas series

Let $L_n$ be the Lucas series: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}(n>1).$ $p$ is a prime number and $p\equiv3,7\pmod {20}$, hence $\exists x,y\in \mathbb Z:2p=x^2+5y^2.$ Is it true that ...
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3answers
55 views

Fibonacci terms with primal distances

Given any prime $p$, are there fibonacci numbers $F_k $ and $F_n$ such that $|F_n - F_k|=p^i , i \in \mathbb N$?
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1answer
149 views

Sums and products involving Fibonacci

In summary, if $\phi$ is the golden ratio, I want to show: \begin{align} \sum_{n=1}^\infty \frac1{F_n} &= 4-\phi \\ \sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{F_nF_{n+1}} &= \phi-1 \\ ...
1
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1answer
59 views

Solving $ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) $?

I need to find $F_{n}$ in : $$ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2 $$ This equation screams convolution , I think , but I find it as a quite long solution sometimes. ...
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3answers
77 views

Find $F_{n}$ in : $F_{n} +2F_{n-1} + … + (n+1)\cdot F_{0} = 3^{n}$

I'm stuck with the question for a while : Find in $$F_{n} +2F_{n-1} + ... + (n+1)\cdot F_{0} = 3^{n}$$ the element $F_{n}$ . Placing $n-1$ instead on $n$ results in : $$F_{n-1} +2F_{n-2} + ... + ...
3
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1answer
81 views

Solve a recursion using generating functions?

Given the recursive equation : $$F_n+F_{n-1}+⋯+F_0=3^n , n\geq0$$ A fast solution that I can think of is placing $n-1$ instead of $n$ , and then we'll get : $$F_{n-1}+F_{n-2}+⋯+F_0=3^{n-1} $$ ...
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4answers
272 views

Fibonacci Cubes: $F_n^3 + F_{n+1}^3 - F_{n-1}^3 =F_{3n}$

Prove $$F_n^3 + F_{n+1}^3 - F_{n-1}^3 =F_{3n}$$ I've tried induction, either its just very long or a neat trick is required in the inductive step but for some odd reason its not working out. ...
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3answers
441 views

Proof of identity $F_m F_n + F_{m−1} F_{n−1} = F_{m+n−1}$ for Fibonacci numbers

I'm lost on where to start on this proof: Using the fact that $A^m A^n = A^{m+n}$ , prove the identity $F_m F_n + F_{m−1} F_{n−1} = F_{m+n−1}$ I want to use induction starting with n = 1, but would ...
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2answers
146 views

Suppose that a recursive routine were invoked to calculate F(4). How many times would a recursive call of F(1) occur?

The definition of a Fibonacci number is as follows: $$F_0=0\\F_1=1\\F_n=F_{n-1}+F_{n-2}\text{ for } n\geq 2$$ Suppose that a recursive routine were invoked to calculate $F_4$. How many times would a ...
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3answers
729 views

Prove that $F(n+3)=2F(n+1)+ F(n)$ for $n \ge 0$

The definition of a Fibonacci number is as follows: $$F(0)=0\\ F(1)=1\\ F(n)= F(n-2)+F(n-1)\text{ for }n\geq 2$$ Prove the given property of the Fibonacci numbers directly from the definition. ...
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1answer
79 views

Is the Fibonacci number-like function too trivial to investigate about?

To make some interesting recursive function, I generalized Fibonacci numbers to a function $f(x)$ such that satisfies the following condition: Given a function $g(x)$, such that $g(0)=0$ and ...
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3answers
1k views

Show that the limit exists and find it's value? [duplicate]

The Fibonacci series defined recursively by $x(1) = 1, x(2) = 2$ and $x(n+1) = x(n) + x(n-1)$ Find$$\lim_{n\rightarrow\infty}\frac{x(n+1)}{x(n)}$$
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4answers
1k views

Is Binet's formula for the Fibonacci numbers exact?

Is Binet's formula for the Fibonacci numbers exact? $F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$ If so, how, given the irrational numbers in it? Thanks.
6
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5answers
298 views

Fibonacci-like sequence

Today I have to deal with something which reminds Fibonacci sequence. Let's say I have a certain number k, which is n-th number of certain sequence. This sequence however is created by recursive ...
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1answer
75 views

What is the broader name for fibonacci and lucas sequences?

Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be: $$x_0 = 53$$ $$x_1 = 62$$ $$x_n ...
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3answers
680 views

What are the first 3 digits of the product of the first 1000 fibonacci numbers

What are the first 3 digits of the product of the first 1000 Fibonacci numbers? Could anyone give me hints on how to start this problem? I haven't done a problem like this before and I am curious ...
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1answer
323 views

Seeking a combinatorial proof of the Fibonacci identity $f_{2k-1}f_{4k}=f_{2k}+f_{2k}f_{4k-1}$

I would appreciate if somebody could help me with the following problem Q: Show that (for $k,n\in \mathbb{N}$), if $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$, then ...
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3answers
95 views

Fiddling with a Fibonacci-Like Sequence

Let $X\in\mathbb{Z}.$ Let $F_n$ be a sequence of positive integers given by $$F_{i+1}=F_i+F_{i-1}$$ $$F_2=X*F_1+F_0$$ I am trying to find an upper bound or (sharp) inequality of $F_i$ in terms of ...
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2answers
131 views

Error in Fibonacci recurrence proof by induction?

I'm working on a problem from a number theory book (Number Theory by George E. Andrews - problem 1-1-11). The text reads: Prove: $\displaystyle F_1F_2+F_2F_3+F_3F_4+\ldots+F_{2n-1}F_{2n}=F_{2n}^2$ ...
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3answers
194 views

Fibonacci sequence: how to prove that $\alpha^n=\alpha\cdot F_n + F_{n-1}$?

Let $F_n$ be the $n$th Fibonacci number. Let $\alpha = \frac{1+\sqrt5}2$ and $\beta =\frac{1-\sqrt5}2$. How to prove that $\alpha^n=\alpha\cdot F_n + F_{n-1}$? I'm completely stuck on this question. ...
3
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1answer
144 views

Have I correctly derived an inverse to the Binet formula?

I was interested by another user's question on finding such an inverse and in particular noted Will Orrick's comment in the best answer that one can square both sides to obtain a quartic.I thought I'd ...
3
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1answer
178 views

Important numbers in Combinatorics

I recently went through some important numbers like the Stirling and Bell number for calculation of partitions /equivalence relations. I was wondering if someone can help me get a list of important ...
3
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1answer
230 views

If $m$ and $n$ are positive integers, then $(F_m,F_n)=F_{(m,n)}$. [duplicate]

Edit: The $F$'s are Fibonacci numbers. I need an idea on how to show the following: If $m$ and $n$ are positive integers, then $(F_m,F_n)=F_{(m,n)}$. I believe that using the fact that ...
3
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1answer
138 views

Invent a combinatorial interpretation for the ''Tribonacci numbers''

" Recall that our combinatorial interpretation of the Fibonacci numbers $f_0 = f_1 = 1$ with $f_n = f_{n-1} + f_{n-2}$ for $n \geq 2$ was the number of ways to tile a board of length $n$ using squares ...
2
votes
3answers
112 views

How to prove? (Do not use mathematical induction)

I would appreciate if somebody could help me with the following problem: Q: Show that $$f_1=f_2=1, f_{n+2}=f_{n+1}+f_{n}(n\in \mathbb{N})~~~~ \Rightarrow ...
0
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2answers
178 views

intuition for the closed form of the fibonacci sequence

I'm trying to picture this closed form from Wikipedia visually: The idea is, if you take $\phi^n / \sqrt{5}$ and round it to the nearest integer, you'll get the $n$th Fibonacci number. I see ...
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votes
8answers
995 views

Need help deriving recurrence relation for even-valued Fibonacci numbers.

That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$ Empirically one can check that: $a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$. If $f(n)$ is ...
5
votes
5answers
394 views

Fibonacci nth term

It is known that the nth term of the Fibonacci sequence can be found by the formula: $F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$, where $\phi$ is the golden ratio (1.618...). Would this be the ...
2
votes
2answers
521 views

How do I prove Binet's Formula? [duplicate]

My initial prompt is as follows: For $F_{0}=1$, $F_{1}=1$, and for $n\geq 1$, $F_{n+1}=F_{n}+F_{n-1}$. Prove for all $n\in \mathbb{N}$: ...
7
votes
1answer
157 views

Identity for $e$ in terms of the Fibonacci sequence.

The following identity appears in Martin Gardner's paper, "Dr. Matrix on Little Known Fibonacci Curiosities: $$e = \frac{1 + 1 + \frac{2}{2!} + \frac{3}{3!} + \frac{5}{4!} + \frac{8}{5!} + ...
3
votes
2answers
96 views

What is the functional inverse (with respect to $h$ (!)) of $f^{\circ h}(x)={F(h) +x F(h-1) \over F(1+h) +x F(h) }$?

I've considered the fractional iteration of $f(x) = {1 \over 1+x} $ for which the general expression depending on the iteration-height parameter $h$ might be assumed by the formula $$ f^{\circ h}(x) ...