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
43 views

Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$

I have the following problem: Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$ Where $F_n$ is the $nth$ Fibonacci number. Proof Basis $n = 6$. $F_6 = 8 \geq 2^{0.5 \cdot ...
3
votes
2answers
60 views

Indexes of prime Fibonacci numbers

I found this on Mathworld, but I can't seem to find any proof, either on StackExchange, nor any other site: Why do all Fibonacci primes, except for $F_4=3$, have prime indexes (with $F_0=0$)? My ...
2
votes
2answers
52 views

Fibonacci Cyclic Pattern [duplicate]

I want to show the Fibonacci numbers are cyclic in mod n. I have tried some small values for n and I can see this is the same. In terms of a proof, I'm thinking of using the pigeonhole principle of ...
2
votes
1answer
47 views

Proving the Fibonacci sum $\sum_{n=1}^{\infty}\left(\frac{F_{n+2}}{F_{n+1}}-\frac{F_{n+3}}{F_{n+2}}\right) = \frac{1}{\phi^2}$ and its friends

In this article, (eq.92) has, $$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_{n+1}F_{n+2}} = \frac{1}{\phi^2}\tag1$$ and I wondered if this could be generalized to the tribonacci numbers. It seems it can ...
4
votes
2answers
142 views

Some infinite series with Fibonacci numbers

An interesting problem is to prove that: $$ \sum_{n=1}^\infty \frac{F_{2n}}{n^2 \binom{2n}{n}}=\frac{4\pi^2}{25 \sqrt 5}. $$ I know the proof, which uses the fact that ...
4
votes
0answers
72 views

On a unique(?) binomial property of $3003$

Given the triangular number, $$T_k = \frac{k(k+1)}{2}$$ and remembering that, $$\binom{n}{m}=\binom{n}{n-m}$$ Excluding $a_0=1$, we then have the six-fold (at least) equalities, $$\begin{aligned} ...
-1
votes
0answers
37 views

Solving Problem 2 in Project Euler using Excel [closed]

I used Excel to answer this (although I first made a mistake). Here's the solution first, followed by a step-by-step methodology https://goo.gl/4EncUo Step 1: Get Fibonacci series formula working ...
0
votes
2answers
39 views

Expression for negating every other odd number index

Is there a way to have an iterative expression that negates every other odd number index (starting from 3)? Basically, I am trying to write a generative expression that will give me value, given ...
2
votes
1answer
77 views

Why is the sum of any ten consecutive Fibonacci numbers always divisible by $11$?

I was wondering if anyone has any insights regarding the fact that the sum of any $a_1, \dots, a_{10}$ consecutive Fibonacci numbers is divisible by $11$ (and furthermore equals to $a_7*11$). What can ...
0
votes
1answer
56 views

Are there ways to separate the Fibonacci sequence? [closed]

I am wondering if there are ways to separate the set of Fibonacci numbers, $F$, into sets $A$ and $B$ such that $$A + B = \{a+b:a\in A,\,b\in B\}=F$$ and such that $A$ and $B$ do not follow the ...
3
votes
2answers
88 views

How do you determine if a number is a even Fibonacci number or not? [duplicate]

Rather than computing out the whole Fibonacci sequence and check if $n$ is even and in there, is there a more straightforward way to compute if $n$ is a even Fibonacci number?
25
votes
1answer
289 views

Fibonacci $\equiv -1 \mod p^2$

Is there a prime $p > 3$ such that the Fibonacci number $F_{np} \equiv -1 \mod p^2$ for some natural number $n$? I know none of the first $1000$ primes $> 3$ qualify. EDIT: In response to ...
6
votes
3answers
100 views

Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$

Given the Fibonacci, tribonacci, and tetranacci numbers, $$F_n = 0,1,1,2,3,5,8\dots$$ $$T_n = 0, 1, 1, 2, 4, 7, 13, 24,\dots$$ $$U_n = 0, 1, 1, 2, 4, 8, 15, 29, \dots$$ and so on, how do we show ...
3
votes
1answer
67 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
2
votes
1answer
38 views

Fibonacci and Lucas numbers congruence relation?

The wikipedia page for Lucas Numbers seems to suggest that if $F_n ≥ 5$ is a Fibonacci number then no Lucas number is divisible by $F_n$. Here is the link. However, the page does not give any ...
1
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0answers
51 views

Proving that the g.c.d of non-consectuive Fibonnaci numbers is also a Fibonacci number

I'm trying to prove that: for non-consecutive Fibonacci numbers, and I know that consecutive Fibonacci numbers are co prime, but I just don't how to prove this using what I know. **EDIT: Lulu has ...
1
vote
1answer
37 views

Cycles in the Fibonacci Sequence mod n with matrices

I was just looking at this question about Fibonacci sequence cycles modulo 5, and I happened to see a very nice solution that involved using matrices. Using the matrix representation of the Fibonacci ...
37
votes
13answers
5k views

How to prove that the Fibonacci sequence is periodic mod 5 without using induction?

The sequence $(F_{n})$ of Fibonacci numbers is defined by the recurrence relation $$F_{n}=F_{n-1}+F_{n-2}$$ for all $n \geq 2$ with $F_{0} := 0$ and $F_{1} :=1$. Without mathematical induction, ...
3
votes
0answers
64 views

Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ...
1
vote
2answers
112 views

Alternating sum of product of Fibonacci numbers

Suppose that $\{F_n\}$ is the sequence of Fibonacci numbers. There is a well-known result that $$\sum_{i=1}^nF_i^2 F_{i+1}=\frac{1}{2}F_nF_{n+1}F_{n+2}.$$ This is easy to prove by induction. I was ...
7
votes
1answer
74 views

Is there a name for this Fibonacci Identity

Last night I was trying to solve a problem and discovered an identity relating to the Fibonacci sequence $$ \left\lvert F_{i-j}F_{i+j} - F_{i-k}F_{i+k} \right\lvert = \left\lvert F_{k - j}F_{k+j} ...
6
votes
4answers
612 views

Last 10 digits of the billionth fibonacci number?

I want to compute the last ten digits of the billionth fibonacci number, but my notebook doesn't even have the power to calculate such big numbers, so I though of a very simple trick: The carry of ...
0
votes
0answers
54 views

Fibonacci series generated with division $1/999999999999999999999998999999999999999999999999$ [duplicate]

I tested the suggestion of finding the Fibonacci series by division, which sounded very surprising to me. I therefore used a simple sympy script to test it and found that it works as advertised. ...
0
votes
2answers
44 views

Proving a Recursive Formula

I know there are some questions on this site about how to find a recursive formula, but I've already found the formula. I'm doing an assignment (http://mathstat.dal.ca/~svenjah/math2112/Assign6.pdf) ...
4
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0answers
39 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
3
votes
1answer
45 views

Prove equality of Fibonacci sequence

Let $u(n)$ — the Fibonacci sequence. Prove that $$u(1)^3+...+u(n)^3=\frac{ 1 }{ 10 } \left[ u(3n+2)+(-1)^{n+1}6u(n-1)+5 \right].$$ I suppose we need prove that equality by iduction in $n$. ...
1
vote
2answers
36 views

proof by induction for golden ratio and fibonacci sequence

I have to prove the following equation by induction for $$x = \phi$$ I am stuck and I don't know how to proceed. This is the equation $$ \phi ^n = f_n\phi + f_{n-1} $$ where $f_n$ is the nth term ...
1
vote
2answers
34 views

Find $r$, given that $F_r= 2F_{101}+F_{100}$

Find $r$, given that $F_r= 2F_{101}+F_{100}$. We know that the recurrence relation for the Fibonacci sequence is $F_n= F_{n-1}+F_{n-2}$ and that $F_0 = F_1 = 1$, but how to proceed further?
3
votes
3answers
112 views

Identity on Fibonacci numbers: $F_{2n}^2=F_{2n+2}F_{2n-2}+1$?

Let $F_n$ be the Fibonacci Sequence ($F_1=F_2=1, F_{n+2}=F_{n+1}+F_{n}$). Prove that $F_{2n}^2=F_{2n+2}F_{2n-2}+1$. I've tried everything from induction to telescoping series but I haven't got close. ...
4
votes
0answers
56 views

Finding first n so nth fibonacci is c modulo p

This is a question I stumbled upon in an online programming contest archive. The problem statement is simple, given $c \equiv F(n)$ mod $P$ and $P$, where $P$ is a prime of form 5$k$ + 1 or 5$k$ - 1, ...
1
vote
2answers
35 views

Fibonacci numbers relation

I was wondering if there was a relation between a Fibonacci number and its position. Is there a function $f(n)$ such that $$f(n)=F_n$$ where $F_n$ is the nth Fibonacci number?
1
vote
1answer
94 views

Proving formulas with products of Fibonacci numbers

While digging through my old notes, I stumbled upon some formulas involving multiplication of Fibonacci numbers that I discovered about 7 years ago (being fascinated with Fibonacci numbers at the ...
0
votes
1answer
44 views

How to find a formula relating fibonacci sequence?

By shifting property of fibonacci numbers, $$F_{m+n} = F_m · F_{n+1} + F_{m-1} · F_n$$ where $F_k$ denotes the kth Fibonacci number . I want to extend it to some n numbers . So , how to find a ...
2
votes
2answers
47 views

Generating Functions with Fibonacci

a) Let \begin{align*} F_{\text{even}}(x) &= F_0x^0 + F_2x^2 + F_4x^4 + F_6x^6 + F_8x^8 + \cdots \\ &= x^2 + 3x^4 + 8x^6 + 21x^8 + \cdots \end{align*} be the generating function whose ...
3
votes
1answer
76 views

Multiple of $p$ in first $p+1$ Fibonacci Numbers

Defining $F_0 = F_1 = 1$ and $F_{n+1} = F_{n} + F_{n-1}$ for $n>0$ gives the Fibonacci sequence, and it is well-known that modulo $p$, one of the first $p+1$ terms is $0.$ In fact, more is known, ...
3
votes
1answer
78 views

Dominos ($2 \times 1$ on $2 \times n$ and on $3 \times 2n$)

How many ways are there to tile dominos (with size $2 × 1$) on a grid of $2 × n$? How about on a grid of $3 × 2n$?
1
vote
1answer
101 views

Is there a pattern of the length between one even Fibonacci number and another?

I had seen a math problem asking for the sum of all even Fibonacci numbers up to 4 million, but I still need to know this: Is there an obvious pattern of the distance between a even Fibonacci number ...
1
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2answers
47 views

Why does Cassini's identity imply consecutive Fibonacci's number are relatively prime?

From Knuth's The Art of Computer Programming, Volume 1 on page 81 he gives Cassini's identity $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. Relation(4) and follows by saying "Relation(4) shows that $F_n$ ...
6
votes
1answer
92 views

Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
13
votes
4answers
306 views

How to prove $ \sum_{k=0}^n \frac{(-1)^{n+k}{n+k\choose n-k}}{2k+1}=\frac{-2\cos\left(\frac{2(n-1)\pi}{3}\right)}{2n+1}$

How to prove $$\sum_{k=0}^n \binom{n+k}{n-k}\frac{(-1)^{n+k}}{2k+1}=-\frac{2}{2n+1}\,\cos\left(\frac{2(n-1)\pi}{3}\right)\;\text{?}$$ I have a proof by induction for it, but it isn't simple! I want ...
2
votes
1answer
27 views

Is this Bertrand's postulate-related statement valid?

Bertrand's postulate says: For every $n>1$ there is always at least one prime $p$ such that $n<p<2n$. Is the following statement: For every $n>3$ there is always at least one ...
45
votes
4answers
4k views

Eyebrow-raising pattern of number of primes between terms of the Fibonacci number sequence?

So, $$1,1,2,3,5,8,13,21...$$ Any connection to primes?...it appears not. However, in between the Fibonacci numbers are how much primes? Let's see: 1 and 1 ZERO 1 and 2 NADA 2 and 3 ZILCH 3 and 5 ZIP ...
1
vote
2answers
71 views

How to solve the difference equation $u_n = u_{n-1} + u_{n-2}+1$

Given that: $$ \begin{equation} u_n=\begin{cases} 1, & \text{if $0\leq n\leq1$}\\ u_{n-1} + u_{n-2}+1, & \text{if $n>1$} \end{cases} \end{equation} $$ How do you solve this ...
4
votes
1answer
40 views

How to determine the generating function?

So I have $$\overset{*}{F} = \overset{*}{F}_{n-1} + \overset{*}{F}_{n-2} + g(n)$$ where $\overset{*}{F}$ is NOT a Fibonacci number for $n \geq 2$. $g(n)$ is any function $g: \mathbb{N} \to ...
2
votes
0answers
40 views

Fibonacci-related infinite sum

Prove that $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}}=\frac{\sqrt{5}}{4}\theta_2^2\bigg(\frac{3-\sqrt{5}}{2}\bigg)$$ and $$\sum_{n=0}^\infty \frac{1}{F_{2n+1}+F_{2k-1}}=\frac{(2k-1)\sqrt{5}}{2F_{2k-1}}$$ ...
4
votes
1answer
43 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
0
votes
1answer
23 views

Proving $F_n \ge (\frac{1}{2}(1+\sqrt{5}))^{n-2}$ for $n \in \mathbb{N}_{>1}$ when $F_n$ is the nth Fibonacci number

Let $F_n$ be defined as the nth Fibonacci number. Prove that $F_n \ge (\frac{1}{2}(1+\sqrt{5}))^{n-2}$ with $n \in \mathbb{N}_{>1}$ My approach thus far was to use induction over $n$. ...
0
votes
3answers
46 views

Prove $F(n) < 2^n$ [closed]

Consider the Fibonacci function $\large{F(n)}$, which is defined such that $F(1) = 1, F(2) = 1$, and $F(n) = F(n−2)+F(n−1)$ for $n > 2$ I know that I should do it using mathematical induction but ...
1
vote
1answer
33 views

An equation to prove with terms of Fibonacci sequence

I would like to prove an equation but I have stuck. The equation that is to prove is the below: $f(n)^2 + (-1)^{n+1} = f(n+1)f(n-1) , n \ge 2$. I'm trying to do an inductive proof of this equation. ...
1
vote
0answers
39 views

Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the ...