Tagged Questions

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

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1answer
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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 $$S(x)-S(x^{2})=x$$ ...
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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 ...
4answers
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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 ...
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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 ...
1answer
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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. ...
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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 ...
4answers
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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: ...
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1answer
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How does one solve this recurrence relation? [closed]

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 ...
3answers
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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 ...
5answers
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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. ...
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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 ...
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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 ...
4answers
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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 ...
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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 ...
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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 ...
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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 ...
2answers
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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 ...
6answers
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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 ...
5answers
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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 Solutions....
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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 ...