Questions regarding functions defined recursively, such as the Fibonacci sequence.

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64
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3answers
2k views

How prove this nice limit $\lim_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$

Nice problem: Let $a_{0}=1$ and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$ Show that ...
39
votes
5answers
2k views

How much weight is on each person in a human pyramid?

After participating in a human pyramid and learning that it's very uncomfortable to have a lot of weight on your back, I figured I'd try to write out a recurrence relation for the total amount of ...
35
votes
0answers
616 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ ...
34
votes
6answers
6k views

Proof the Fibonacci numbers are not a polynomial.

I was asked a while ago to prove there is no polynomial $P$ in $\mathbb R$ such that $P(i)=f_i$ for all $i\geq0$. I tried to get a proof as slick as possible and here's what I got. Let ...
33
votes
1answer
927 views

Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$

If $S = x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Find S. Note:This is not a GP series.The powers are in GP. My Attempts so far: 1)If $S(x)=x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$ Then ...
33
votes
2answers
721 views

Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$

In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
33
votes
1answer
1k views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
32
votes
4answers
2k views

A stronger version of discrete “Liouville's theorem”

If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, ...
32
votes
3answers
931 views

Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
31
votes
3answers
794 views

Interesting representation of $e^x$

So I discovered the following formula by using the Taylor series for $\ln (x+1)$ $$x= \ln ...
28
votes
2answers
1k views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...
28
votes
2answers
381 views

Find $\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+…}}}}$

Find the value of $$\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+...}}}}$$ I know how to solve when all surds are of the same order, but what if they are different? Technically, (as some users ...
27
votes
2answers
429 views

$ \sqrt{2} \sqrt{2 - \sqrt{2}} \sqrt{2 - \sqrt{2 - \sqrt{2}}} \sqrt{2 - \sqrt{2 - \sqrt{2-\sqrt{2}}}} \cdots $ =?

In other words, if we define a sequence $$ \displaystyle a_{n+1} = \sqrt{2-a_n}, \,\,\,a_0 = 0 .$$ Then, we need to find $$ \displaystyle \prod_{n=1}^{\infty}{a_n}. $$ Well, from here I don't seem ...
27
votes
2answers
571 views

How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?

I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$, where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$. Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...
26
votes
3answers
571 views

Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following conjecture true? A conjecture : If $a_n$ is an integer, then $n\le 8$. I ...
25
votes
2answers
583 views

Is $\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin x\cdots\right)\right)=\frac4{\pi}\sum\limits_{k=0}^\infty\frac{\sin(2k+1)x}{2k+1}$?

We can see intuitively that $$ f(x)=\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin{x}\cdots\right)\right)\right) $$ is the square wave with period $2\pi$ and has the ...
24
votes
1answer
278 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
23
votes
1answer
4k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
22
votes
4answers
632 views

Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$

Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical $$ \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}} ...
20
votes
5answers
1k views

Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$

Today, we had a math class, where we had to show, that $a_{100} > 14$ for $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$ Apart from this task, I asked myself: Is there a closed form for this ...
20
votes
3answers
1k views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

I am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all ...
20
votes
6answers
822 views

Why does every “fibonacci like” series converge to $\phi$?

It's is well known that the ratio of side-by-side fibonacci numbers converge to $\phi$. But it seems by my calculations, that if one starts with any pair of numbers one will also get a ratio that ...
20
votes
4answers
489 views

recurrence relation arising from Magic the Gathering scenario [duplicate]

Possible Duplicate: Probability of a random binary string containing a long run of 1s? EDIT: Cocopuffs below has partially answered the question, but the critical base case $L=2$ to his ...
20
votes
2answers
1k views

Probability of a random binary string containing a long run of 1s?

For some fixed $n$, let $p_n$ be the probability that a random infinite binary string contains a run of consecutive $1$s, containing $n$ more $1$s than the total number appearing before the run. For ...
20
votes
1answer
487 views

Infinite staircase to a circle

Suppose you start at $(0,0)$ on the unit disc and repeat the following procedure again and again: Face east and walk half-way to the circumference. Face north and walk half-way to the circumference. ...
20
votes
2answers
444 views

What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve

What is $$\int_{K} e^{a \cdot x+ b \cdot y} \mu(x,y)$$ where $K$ is the Koch curve and $\mu(x,y)$ is a uniform measure look here. Attempt: I can evaluate the integral numerically and I have derived ...
19
votes
2answers
891 views

Self-avoiding walk on $\mathbb{Z}$

How many sequences $a_1,a_2,a_3,\dotsc$, satisfy: i) $a_1=0$ ii) ($a_{n+1}=a_n-n$ or $a_{n+1}=a_n+n$) iii) $a_i\neq a_j$ for $i\neq j$ iiii) $\mathbb{Z}=\{a_i\}_{i>0}$ Are the two alternating ...
18
votes
4answers
4k views

Number of ways to partition a rectangle into n sub-rectangles

How many ways can a rectangle be partitioned by either vertical or horizontal lines into n sub-rectangles? At first I thought it would be: ...
18
votes
3answers
620 views

Counting binary sequences with no more than $2$ equal consecutive numbers

I invented the following problem, but I can't solve it. There are $n$ $1$'s and $n$ $0$'s and my question is what is the number of ways to arrange them avoiding $3$ equal consecutive numbers. So for ...
18
votes
3answers
367 views

Recurrence relations and limits, tough.

I would like a hint for the following, more specifically, what strategy or approach should I take to prove the following? Problem: Let $P \geq 2$ be an integer. Define the recurrence $$p_n = p_{n-1} ...
18
votes
1answer
298 views

Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: ...
17
votes
2answers
2k views

Explicit formula for Bernoulli numbers by using only the recurrence relation

It is not hard to show, by induction on $m\in\mathbb N$, that there exist a sequence $(B_n)_{n\geq0}$ of rational numbers such that ...
16
votes
3answers
911 views

Solving the recursion $3a_{n+1}=2(n+1)a_n+5(n+1)!$ via generating functions

I have been trying to solve the recurrence: \begin{align*} a_{n+1}=\frac{2(n+1)a_n+5((n+1)!)}{3}, \end{align*} where $a_0=5$, via generating functions with little success. My progress until now is ...
16
votes
5answers
884 views

limit of : $a_{n+2} =\frac{1}{a_n} + \frac{1}{a_{n+1}}$

$a_n$ is a real sequence, $a_1,a_2$ are positive and for all $n>2$ : $$ a_{n+2} =\frac{1}{a_{n+1}} + \frac{1}{a_{n}}.$$ Prove that: $\displaystyle \lim_{n\to \infty} a_n$ exists, then find it. ...
16
votes
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 ...
16
votes
2answers
347 views

Generalized Fibonacci Sequence Question

The Fibonacci Sequence is defined as the recurrence $a(n)=a_{n-1}+a_{n-2}$ where $a(0)=0$ and $a(1)=1$. Today, I was bored so I considered the sequence $a(n)=\sqrt{a_{n-1}}+\sqrt{a_{n-2}}$. Ten ...
15
votes
4answers
965 views

Finding an Explicit Formula from the Recurrence: $na_{n}= 2 ( a_{n-1}+a_{n-2})$

Here is the recurrence: $$na_{n}=2(a_{n-1}+a_{n-2}) \qquad\text{ where } a_{0}=1\text{ and }a_{1}=2$$ At first I thought that this could be easily solved by simply multiplying the Fibonacci ...
15
votes
4answers
342 views

Solve recursion $a_{n}=ba_{n-1}+cd^{n-1}$

Let $b,c,d\in\mathbb{R}$ be constants with $b\neq d$. Let $$\begin{eqnarray} a_{n} &=& ba_{n-1}+cd^{n-1} \end{eqnarray}$$ be a sequence for $n \geq 1$ with $a_{0}=0$. I want to find a closed ...
15
votes
2answers
270 views

How to prove that $a_{n}$ must be of the form $a^2+b^2$?

let $a_{1}=1,a_{2}=2,a_{3}=5$,and $$a_{n}=3a_{n-1}a_{n-2}-a_{n-3}$$ show that $a_{n}=a^2+b^2,a,b\in N$ while $a_{1}=0^2+1^2,a_{2}=2=1^2+1^2,a_{3}=5=2^2+1^2,a_{4}=29=5^2+2^2,a_{5}=433=17^2+12^2$ and ...
15
votes
2answers
588 views

Solving the recurrence relation $a_n=\frac{a_{n-1}^2+a_{n-2}^2}{a_{n-1}+a_{n-2}}$

I would like to know if there is a way to get the recurrence relation $$a_n=\frac{a_{n-1}^2+a_{n-2}^2}{a_{n-1}+a_{n-2}},\qquad (a_1=1,a_2=2)$$ in closed form, or if there is no such way, how one could ...
14
votes
4answers
892 views

How does one solve this recurrence relation? [on hold]

We have the following recursive system: $$ \begin{cases} & a_{n+1}=-2a_n -4b_n\\ & b_{n+1}=4a_n +6b_n\\ & a_0=1, b_0=0 \end{cases} $$ and the 2005 mid-exam wants me to calculate answer ...
14
votes
6answers
2k views

How to solve this recurrence relation? $f_n = 3f_{n-1} + 12(-1)^n$

How to solve this particular recurrence relation ? $$f_n = 3f_{n-1} + 12(-1)^n,\quad f_1 = 0$$ such that $f_2 = 12, f_3 = 24$ and so on. I tried out a lot but due to $(-1)^n$ I am not able to ...
14
votes
5answers
473 views

Alternate proof that for every natural number $n,\ \left\lfloor\left(\frac{7+\sqrt{37}}{2}\right)^n\right\rfloor$ is divisible by $3$

Original Problem: Prove that for every natural number $n$,$$\left\lfloor\left(\frac{7+\sqrt{37}}{2}\right)^n\right\rfloor$$ is divisible by $3$. I found the problem in the book Winning ...
14
votes
3answers
5k views

Solving recurrence relation in 2 variables

We already know how to solve a homogeneous recurrence relation in one variable using characteristic equation. Does a similar technique exists for solving a homogeneous recurrence relation in 2 ...
14
votes
4answers
519 views

A recurrence relation for the Harmonic numbers of the form $H_n = \sum\limits_{k=1}^{n-1}f(k,n)H_k$

Working on Harmonic numbers, I found this very interesting recurrence relation : $$ H_n = \frac{n+1}{n-1} \sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{1+n-k}\right)H_k ,\quad \forall\ ...
14
votes
1answer
2k views

Solving a Nonlinear Recursion

In the course of some research computations I have been doing, I run up against a recursion $$ a_{n+3} = a_{n+2}a_{n+1} - a_n $$ I've tried to find out if it's possible to solve recursions of this ...
13
votes
5answers
470 views

Solve the sequence : $u_n = 1-(\frac{u_1}{n} + \frac{u_2}{n-1} + \ldots + \frac{u_{n-1}}{2})$

For a physical model I am trying to solve this sequence: $$\begin{align*} u_1 &= 1 \\ u_2 &= 1-\left(\frac{u_1}{2}\right) \\ u_3 &= 1-\left(\frac{u_1}{3} + \frac{u_2}{2}\right) \\ u_4 ...
13
votes
3answers
256 views

For $x_{n+1}=x_n^2-2$, show $\lim_{n\to\infty}\frac{x_n}{x_0x_1\cdots x_{n-1}}=2$

Suppose $x_0:=2\sqrt{2}$ and $x_{n+1}=x_n^2-2$ for $n\ge1$. We have to show $$\lim_{n\to\infty}\frac{x_n}{x_0x_1\cdots x_{n-1}}=2$$ Establishing convergence is pretty direct but I'm having trouble ...
13
votes
3answers
393 views

Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
13
votes
3answers
1k views

Recursive Integration over Piecewise Polynomials: Closed form?

Is there a closed form to the following recursive integration? $$ f_0(x) = \begin{cases} 1/2 & |x|<1 \\ 0 & |x|\geq1 \end{cases} \\ f_n(x) = ...