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

learn more… | top users | synonyms (2)

6
votes
1answer
173 views

permutation and f(n) challenge

Suppose $f(n)$ be the number of permutation from set ${1,2,..,n}$ such that for each $ 1 \leq i \leq n$ we have: $ | \pi(i)-i| \leq 1 $. meaning of $ \pi(i)$ is an elements whose in place $i$ of ...
6
votes
2answers
139 views

$a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$

Define a sequence $\left\lbrace a_{n}\right\rbrace$ by $\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor ...
6
votes
1answer
140 views

How to prove that this recursively defined sequence converges to $e$?

Let $a_1=0,a_2=1,$ and $a_{n+2}=\dfrac{(n+2) a_{n+1}-a_n}{n+1}$. Prove that $\lim_{n\to \infty}a_n=e$. I know that $\lim_{n\to\infty}\left(1+\frac1{2!}+\frac1{3!}+...+\frac1{n!}\right)=e$ and $a_n = ...
6
votes
1answer
480 views

Recurrence relation telescoping

Hi there I am trying to solve the following recurrence relation using telescoping. How would I go about doing it? $$T(n) = \frac 2n \Big(T(0) + T(1) + \ldots+ T(n-1)\Big) + 5n$$ Assuming $n\ge 1$
6
votes
1answer
557 views

Towers of Hanoi - are there configurations of $n$ disks that are more than $2^n - 1$ moves apart?

This is an exercise from Chapter 1 of "Concrete Mathematics". It concerns the Towers of Hanoi. Are there any starting and ending configurations of $n$ disks on three pegs that are more than $2^n - ...
6
votes
2answers
124 views

Count with only certain digits allowed - And feel a fractal

I have a friend ~200 years old mathematician who has forgotten some digits and now he counts things in very strange manner: when he is going to count the $n$-th thing and $n$ contains a digit he ...
6
votes
1answer
191 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
6
votes
2answers
197 views

Generating Functions: Solving a Second-Order Recurrence

I'm self-studying generating functions (using GeneratingFunctionology as a text). I came across this programming problem, which I immediately recognized as a modification of the Fibonacci sequence. ...
6
votes
2answers
92 views

Recurrence for expected length of Gaussian vector

Let $g_k \sim N(0, I_{k \times k})$ be a a standard $k$-dimensional Gaussian vector. Denote by $\|g\|$ the $2$-norm of $g$. By explicit integration, it is not hard to see that $$ \mathbb E \|g_k\| = ...
6
votes
1answer
132 views

Limit related to the recursion $a_{n+1}=(1-n^{-1/4}a_n)a_n$

Let $0<a_{1}<1$, with $$a_{n+1}=a_{n}(1-n^{-\frac{1}{4}}a_{n})$$ Does there a real numbers $A$ which make the limit $$ ...
6
votes
2answers
196 views

The limit of a recurrence relation (with resistors)

Background to problem (not too important): My proposed solution: The infinitely long element, , however complex, can be represented as a single resistor of resistance $R$. Remembering the ...
6
votes
1answer
309 views

Finding ($2012$th term of the sequence) $\pmod {2012}$

Let $a_n$ be a sequence given by formula: $a_1=1\\a_2=2012\\a_{n+2}=a_{n+1}+a_{n}$ find the value: $a_{2012}\pmod{2012}$ So, in fact, we have to find the value of ...
6
votes
1answer
351 views

Finding an explicit formula from a recursive formula.

I have the recurrence relation: $$g(k, 0, x) = k,$$ $$g(k, n, x) = \dfrac{1}{2} \log_{k}{\left(\dfrac{k^{g(k, n - 1, x)}x}{g(k, n - 1, x)}\right)},$$ and I would like to solve it, if it is ...
6
votes
4answers
64 views

Proving that the elements of $A^n$ matrix are non-zero

Let $ A = \begin{pmatrix} 1 & 2 \\-1 & 1 \\ \end{pmatrix}$. Prove that for every positive integer $n$ there exist integers $x_{n},y_{n}$ such that $A^n= \begin{pmatrix} x_{n} & -2y_{n} \\ ...
6
votes
1answer
136 views

How find $a_{n}$ if $a_{n+1}=\sqrt{2a_n+1}$

let $a_{1}=\dfrac{\sqrt{2}}{4}$ and such $$a_{n+1}=\sqrt{2a_{n}+1}$$ find $a_{n}$ my idea:let $a_{n}=\dfrac{1}{2}\cos{x_{n}}$ $$\Longrightarrow ...
6
votes
1answer
194 views

Line in a proof on p69 in Cassel's Local Fields

I'm trying to read the proof of LEMMA 6.1 (Nagell) Let $u_n$ be defined by $u_0=0$, $u_1=1$ and $u_n=u_{n-1}-2u_{n-2} \hspace{20pt} (n\geq 2)$. Then $u_n=\pm1$ only for $n=1,2,3, ...
6
votes
1answer
91 views

How can I find $f(3)$ if $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$?

If $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$ and $f(1) = 2$ then find $f(3)$. Can you give me any hint that I can start with?
6
votes
1answer
83 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
6
votes
1answer
123 views

A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$

I am not sure whether this is right. Can anyone verify, whether this proof is valid? Thanks! Define a sequence $\{a_n\}_{n\ge0}$ as follows: ...
6
votes
2answers
379 views

How to prove $\lim_{n\to\infty}\frac{\log_{2}{(f_{n})}}{(\log_{2}{n})^2}=\frac{1}{2}$?

For any $n\in N$, such $f_{1}=1$, and such $$f_{2n+1}=f_{2n}=f_{2n-1}+f_{n},$$ prove that $$\lim_{n\to\infty}\dfrac{\log_{2}{(f_{n})}}{(\log_{2}{n})^2}=\dfrac{1}{2}.$$
6
votes
1answer
195 views

Recurrence Relation

How do I solve: $k(k+1)a_{k}=2(\lambda k-1)a_{k-1}+(a-\lambda^2)a_{k-2}$ where $\lambda$ and $a$ are constants, and similar other recurrence relations?
6
votes
2answers
2k views

Solving recurrence $T(n) = T(\lceil n/2 \rceil) + T(\lfloor n/2 \rfloor) + \Theta(n)$

I'm learning algorithms by myself and am using the excellent Introduction to algorithms to do that. It has been quite a long time since I last studied math, so maybe the solution to my problem is ...
6
votes
1answer
56 views

The number of series over $\{0,1,2\}$ without repeating numbers

What is the number of series over $\{0,1,2\}$ with length $n$ without repeating the same number one after the other ($22$ is not allowed but $101$ is), that does not begin and end with the number $2$. ...
6
votes
1answer
1k views

Combinatorics - Catalan numbers recursive function proof by induction help

I'm trying to prove that $$C_n = \sum_{i=0}^{n-1} C_iC_{n-i-1}$$ when $1\leq n$ and $C_0 = 1$ by induction. ($C_n$ is the $n$th catalan number). For $n=1$ this is true. Assume it is true for $n=k$: ...
6
votes
1answer
156 views

Prove that every element of $a_{n+2013}=\frac{a_{n+1}a_{n+2}…a_{n+2012}+1}{a_n}$ is an integer

Given $\displaystyle a_1=a_2=\cdots=a_{2013}=1$ and $\displaystyle a_{n+2013}=\frac{a_{n+1}a_{n+2}\cdots a_{n+2012}+1}{a_n}$. Prove that $a_{n+2013}\in\mathbb{N}$ for all $n\in\mathbb{N}$. I ...
6
votes
0answers
273 views

Sum of infinite series defined recursively

Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows: The initial term, $y_1$, is the ...
6
votes
0answers
79 views

Right way to solve this reccurence relation?

I am not that good in mathematics , I have solved many simple relations. But I am not be able to solve the following. This is not a homework. I found it during analysis of a program. ...
6
votes
0answers
738 views

Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims ...
5
votes
5answers
345 views

Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$

$T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $ Show that $T(n) = 2^n$. I know how to prove this by induction, but I would like to know how to show this using first principles. Edit: The way I want ...
5
votes
4answers
313 views

General expression of $f(a, b)$ if $f(a, b)=f(a-1,b) + f(a, b-1) + f(a-1, b-1)$?

$f(a,b) = f(a-1, b) + f(a-1, b-1) + f(a, b-1), ab \neq 0$ $f(a,b) = 1, ab = 0$ So what is $f(a, b)$?
5
votes
4answers
547 views

What explains this bizarre behavior?

Consider the sequence $x_{n+1} = 2x_{n}-\frac{1}{x_n},n\geq0 $. For $x_{0} = 0,87$ we have $$ \begin{aligned} X(1) &\approx 0,590574712643678 \\ X(2) &\approx -0,512116436915835\\ X(3) ...
5
votes
3answers
540 views

How to slice the cheese

I encounter a problem recently stated as below: How many pieces of cheese we can obtain from a single thick piece by making five straight slices?(we can't move the cheese when slicing)If we wanna ...
5
votes
3answers
1k views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
5
votes
2answers
223 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
5
votes
3answers
411 views

How to solve this recurrence using generating functions?

Exercise: For $n \geq 0$ let $a_n = \sum \limits_{i=0}^n (i^2- 2i + 1)$ a) Show that $$a_{n+4} -4a_{n+3} + 6a_{n+2} - 4a_{n+1} + a_n = 0, n \geq 0$$ b) Identify the genereating series ...
5
votes
2answers
442 views

A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of ...
5
votes
3answers
991 views

Analysis of Algorithms: Solving Recursion equations-$T(n)=3T(n-2)+9$

I need your help with solving this recursion equation: $T(n)=3T(n-2)+9$. with the initial condition : $T(1)=T(2)=1$. I need to find $T(n)$, the complexity of the algorithm which works that way. I ...
5
votes
2answers
16k views

How to solve this recurrence $T(n) = 2T(n/2) + n\log n$

How can I solve the recurrence relation $T(n) = 2T(n/2) + n\log n$? It almost matches the Master Theorem except for the $n\log n$ part.
5
votes
5answers
2k views

Solving a set of recurrence relations

I have the 7 following reccurence relations: $A_n = B_{n-1} + C_{n-1}$ $B_n = A_n + C_{n-1}$ $C_n = B_n + C_{n-1}$ $D_n = E_{n-1} + G_{n-1}$ $E_n = D_n + F_{n-1}$ $F_n = G_n + C_n$ $G_n = E_n + ...
5
votes
4answers
474 views

How to find the general formula for this recursive problem?

I want to ask about recursive problem. Given: $$a_0= 11, a_1= -13,$$ and $$a_n= -a_{n-1} +2a_{n-2}.$$ What is the general formula for $$a_n$$ ? I've already tried to find the first terms of this ...
5
votes
2answers
189 views

Finding non-negative integers $m$ such that $(1 + \sqrt{-2})^m$ has real part $\pm 1$.

I believe that the integers $m$ with $(1+\sqrt{-2})^m$ having real part $\pm 1$ are $0, 1, 2$ and $5$, but I'm having trouble proving it. Write $$a_m = \Re((1+\sqrt{-2})^m) = \frac{(1 + \sqrt{-2})^m ...
5
votes
5answers
351 views

What particular solution should I guess for this relation?

Just trying to solve a non-homogeneous recurrence relation: $$f(n) = 2f(n-1) + n2^n$$ $$f(0) = 3$$ Characteristic equation is: $$f(n) - 2f(n-1) = 0$$ $$a-2 = 0$$ $$a = 2$$ Homogeneous ...
5
votes
4answers
755 views

How to solve this recurrence relation?

There's a frog who could climb either 1 stair or 3 stairs in one shot. In how many ways he could reach at 100th stair? I came up with the solution $g(n) = g(n-3) + g(n-1)$, where $g(0)=g(1)=g(2)=1$ ...
5
votes
3answers
142 views

Prove that this recurrence always cycles

If $n$ is a nonnegative integer, let $S_n=\{0, 1, 2, \dots, 2n+1\}$. For $t\in S_n$ repeatedly perform if t is even t = t/2 else t = (n + 1 + ⌊t/2⌋) ...
5
votes
3answers
84 views

sequence $U_k=a_0U_{k-1}+a_1U_{k-2}+\cdots+a_{k-1}U_0$

Is there a general formula for $U_k$ defined by $$U_k=a_0U_{k-1}+a_1U_{k-2}+\cdots+a_{k-1}U_0$$ where the $a_i$ are in arithmetic progression and $U_0=1$? Do there always exist $c,d$ such that ...
5
votes
3answers
2k views

Recurrence trouble: $T(n)=2T(n/2)+T(n/3)+\theta(n^2)$

I have to solve the following recurrence :$\displaystyle T(n)=2T(n/2)+T(n/3)+\theta(n^2)$ I have done the whole tree analyses and now I have to prove that $\displaystyle T(n) \leq ...
5
votes
3answers
296 views

Mathematical Induction proof for a cubic equation.

If $ x^3 = x +1$, prove by induction that $ x^{3n} = a_{n}x + b_n + \frac {c_n}{x}$, where $a_1=1, b_1=1, c_1=0$ and $a_n = a_{n-1} + b_{n-1}, b_n = a_{n-1} + b_{n-1} + c_{n-1}, c_n = a_{n-1} + ...
5
votes
1answer
183 views

Strange Recurrence: What is it asymptotic to?

So I have the following recurrence relation for the growth rate of an algorithm: $T(n)$ = time taken by algorithm to solve problem of size n: $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$ Clearly this ...
5
votes
2answers
304 views

Sum of the series formula

I need to figure out the sum of the series as quickly as possible in a program given n and k: $$f(n,k)= ...
5
votes
3answers
95 views

What are the possible limits of the iteration $x_{n+1}=\sqrt{x_n+3}$, $x_0=0$?

Let $f(x)=\sqrt{x+3}$ for $x\ge -3$. Consider the iteration $$x_{n+1}=f(x_n),x_0=0;n\ge 0$$ The possible limits of the iteration are -1 3 0 $\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}$ I think only ...