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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6
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1answer
66 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
6
votes
3answers
1k views

People sitting in a circle chewing gum

Ten people are sitting in a circle of ten chairs, chewing gum. Each person spits out his or her gum and places it either under his or her own chair or under an immediately adjacent chair. How many ...
2
votes
2answers
50 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
64 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
145 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
73 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
vote
6answers
67 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
78 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
89 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
292 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
vote
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
67 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
613 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
votes
0answers
40 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
57 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
81 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
103 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
vote
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
93 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
35 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. ...