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

learn more… | top users | synonyms (2)

4
votes
1answer
1k views

Solving a recurrence relation with the characteristic equation

I have some trouble solving this due to not seeing the steps to be able to feed it into the characteristic equation. $$T(n) = 4T(n-2) +n + 2^nn^2\ \text{with}\ \ T(0)=0,\ T(1)=1$$ (don't have to ...
6
votes
3answers
4k views

I have a problem understanding the proof of Rencontres numbers (Derangements)

I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation $$D_{n,0}=\left[\frac{n!}{e}\right]$$ where $[\cdot]$ denotes the rounding function (i.e., ...
14
votes
6answers
1k 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 ...
1
vote
3answers
221 views

Can Master Theorem be applied on any of these?

1) $T(n) = 6T(n/2) + 2^{3 \log(n)}$ 2) $T(n) = 8T(n/2) + \frac{n^3}{(\log(n))^4}$ 3) $T(n) = 9T(n/3) + n(\log(n))^3$ Can the complexity for these be calculated with the Master Theorem? I am not sure ...
1
vote
4answers
495 views

Solving a recurrence of polynomials

I am wondering how to solve a recurrence of this type $$p_1(x) = x$$ $$p_2(x) = 1-x^2$$ and $$p_{n+2}(x) = -xp_{n+1}(x)+p_{n}(x).$$ I am wondering, how could one solve such a recurrence. One way ...
27
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, ...
17
votes
5answers
1k views

Find a closed form for this sequence: $a_{n+1} = a_n + a_n^{-1}$

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 my self: Is there a closed form for this ...
5
votes
3answers
358 views

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

How can I solve the recurrence relation $$T(n)=2T\left(\frac n2\right)+\frac{n}{\log n}$$? I am stuck up after few steps.. I arrive till $$T(n) = 2^k T(1) + \sum_{i=0}^{\log(n-1)} ...
6
votes
2answers
237 views

recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
6
votes
2answers
2k views

Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt ...
5
votes
4answers
5k views

What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?

This looks like a question asked earlier, but it isn't. $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad & \text{ if } n>1 \\ 1 & \text{ if }n=1\end{cases}$$ My professor gave this to me in ...
7
votes
2answers
554 views

unorthodox solution of a special case of the master theorem

I am asking for references regarding a special case of the master theorem. This theorem seems to appear quite a lot on this site, prompting me to study it in more detail, e.g. see my posts here and ...
6
votes
1answer
302 views

convoluted recurrence: $f(2n)=f(n)+f(n+1)+n, f(2n+1)=f(n)+f(n-1)+1$

For the recurrence relation: $f(0)=1$ $f(1)=1$ $f(2)=2$ $f(2n)=f(n)+f(n+1)+n\ \ \ (\forall n>1)$ $f(2n+1)=f(n)+f(n-1)+1\ \ \ (\forall n\ge 1)$ (the first numbers of the sequence are: 1, 1, 2, ...
8
votes
4answers
5k views

How to find the closed form formula for this recurrence relation

$ x_{0} = 5 $ $ x_{n} = 2x_{n-1} + 9(5^{n-1})$ I have computed: $x_{0} = 5, x_{1} = 19, x_{2} = 83, x_{3} = 391, x_{4} = 1907$, but cannot see any pattern for the general $n^{th}$ term.
1
vote
4answers
4k views

Solve the recurrence relation:$ T(n) = \sqrt{n} T \left(\sqrt n \right) + n$

$$T(n) = \sqrt{n} T \left(\sqrt n \right) + n$$ Master method does not apply here. Recursion tree goes a long way. Iteration method would be preferable. The answer is $Θ (n \log \log n)$. Can ...
15
votes
2answers
1k 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 ...
5
votes
2answers
7k 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.
3
votes
2answers
173 views

Find the asymptotic tight bound for $T(n) = 4T(n/2) + n^{2}\log n$

Find the asymptotic tight bound in $$ T(n) = 4T\left(\frac{n}{2}\right) + n^{2}\log n. $$ where $ \log n= \log _{2}n $ and $T(1) = 1$. I should solve this using all three common methods: iteration, ...
3
votes
3answers
257 views

Recurrence relation by substitution

I have an exercise where I need to prove by using the substitution method the following $$T(n) = 4T(n/3)+n = \Theta(n^{\log_3 4})$$ using as guess like the one below will fail, I cannot see why, ...
2
votes
1answer
926 views

Recurrence relation, Fibonacci numbers

$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$. $(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the integers $a_n$ and $a_{n+1}$ ...
1
vote
1answer
303 views

Establishing formula from recurrence

Can anyone tell me how do we establish a formula from a given recurrence relation? Take the example of $f(n) = 2f(n-1) + 1$, $n \in \mathbb{Z^+}$, $f(1) = 1$ When I write out the first few values, it ...
8
votes
3answers
1k views

Solving a Recurrence Relation/Equation, is there more than 1 way to solve this?

1) Solve the recurrence relation $$T(n)=\begin{cases} 2T(n-1)+1,&\text{if }n>1\\ 1,&\text{if }n=1\;. \end{cases}$$ 2) Name a problem that also has such a recurrence relation. The ...
4
votes
2answers
2k views

Fibonacci, tribonacci and other similar sequences

I know the sequence called the Fibonacci sequence; it's defined like: $\begin{align*} F_0&=0\\ F_1&=1\\ F_2&=F_0+F_1\\ &\vdots\\ Fn&=F_{n-1} + F_{n-2}\end{align*}$ And we ...
9
votes
4answers
305 views

Help with a recurrence with even and odd terms [duplicate]

I have the following recurrence that I've been pounding on: $$ a(0)=1\\ a(1)=1\\ a(2)=2\\ a(2n)=a(n)+a(n+1)+n\ \ \ (\forall n>1)\\ a(2n+1)=a(n)+a(n-1)+1\ \ \ (\forall n\ge 1) $$ I don't have much ...
2
votes
4answers
343 views

can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer

derive the $n^{th}$ term for the series $0,1,3,7,15,31,63,127,255,\ldots$ observation gives, $t_{n}=2^n-1$, where $n$ is a non-negative integer $t_{0}=0$
11
votes
5answers
888 views

How to deduce a closed formula given an equivalent recursive one?

I know how to prove that a closed formula is equivalent to a recursive one with induction, but what about ways of deducing the closed form initially? For example: $$ f(n) = 2 f(n-1) + 1 $$ I know ...
7
votes
5answers
541 views

How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$

I am self-studying Discrete Mathematics, and I have two exercises to solve. Find a formula for the following recurrence relation: (translated from Portuguese) a) $a_{1}=3,a_{2}=5, ...
3
votes
1answer
334 views
4
votes
1answer
119 views

Recurrence $a_n = \sum_{k=1}^{n-1}a^2_{k}, a_1=1$

This seems like a really straightforward recurrence. I wrote out the first few terms: $1,1,2,6,42,1806$... It seems to grow faster than $n!$ but slower than $n^n$. Any suggestions about the closed ...
3
votes
2answers
496 views

We define a sequence of rational numbers {$a_n$} by putting $a_1=3$ and $a_{n+1}=4-\frac{2}{a_n}$ for all natural numbers. Put $\alpha = 2 + \sqrt{2}$

We define a sequence of rational numbers {$a_n$} by putting $a_1=3$ and $a_{n+1}=4-\frac{2}{a_n}$ for all $n \in \mathbb N$. Put $\alpha = 2 + \sqrt{2}$ (a) Prove by induction on n, that $3 \le a_n ...
1
vote
2answers
160 views

Solving the recurrence $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{2}\log(n)$

Please help me solve the recurrence $$ T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{2}\log(n) $$
1
vote
2answers
259 views

Finding the asymptotic behavior of the recurrence $T(n)=4T(\frac{n}{2})+n^2$ by using substitution method

I am trying to solve a recurrence by using substitution method. The recurrence relation is: $$T(n)=4T\left(\frac{n}{2}\right)+n^2$$ My guess is $T(n)$ is $\Theta (n\log n)$ (and I am sure about it ...
0
votes
2answers
3k views

Recurrence telescoping $T(n) = T(n-1) + 1/n$ and $T(n) = T(n-1) + \log n$

I am trying to solve the following recurrence relations using telescoping. How would I go about doing it? $T(n) = T(n-1) + 1/n$ $T(n) = T(n-1) + \log n$ thanks
58
votes
3answers
1k 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 ...
16
votes
3answers
811 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 ...
13
votes
3answers
227 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
5answers
440 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 ...
7
votes
2answers
1k views

Proof of clockwise towers of Hanoi variant recursive solution

This is from one of the exercises in "Concrete Mathematics", and is something I'm doing privately, not homework. This is a variant on the classic towers of Hanoi, where all moves must be made ...
1
vote
1answer
183 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
4
votes
1answer
57 views

Simplying linear recurrence sum with binomials

Is there a way to simplify $$\sum_{k=1}^{n} \binom{n}{k}f(k)$$ Where $f(k)$ is a large linear recurrence?
4
votes
3answers
2k views

Recurrence relation (linear, second-order, constant coefficients)

Q1. Find the general solution to the difference equation $$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$ Q2. Solve the difference equation $$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$ I am ...
3
votes
1answer
226 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
3
votes
3answers
652 views

expansion of $\cos^k(\theta)$

Does any body know a expansion of : $\cos^k(\theta)$ in function of $\cos$ and/or $\sin$ but without power? For example : $\cos^2(\theta)=\frac{1}{2}(\cos(2\theta)+1)$, but i would want a ...
2
votes
5answers
5k views

Solve the recurrence $T(n) = 2T(n-1) + n$

Solve the recurrence $T(n) = 2T(n-1) + n$ where $T(1) = 1$ and $n\ge 2$. The final answer is $2^{n+1}-n-2$ Can anyone arrive at the solution?
10
votes
2answers
2k views

Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct?

Problem The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation: $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Evaluate the limit $\lim_{n\to\infty} a_n$. Solution Show that the sequence ...
7
votes
5answers
403 views

The sum $\sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j$

A recent answer of mine to a question on Math Overflow includes the sum $$S(n,k,x) = \sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j,$$ where $\left\{ j \atop k \right\}$ is a Stirling number ...
5
votes
5answers
126 views

Recurrence of the form $2f(n) = f(n+1)+f(n-1)+3$

Can anyone suggest a shortcut to solving recurrences of the form, for example: $2f(n) = f(n+1)+f(n-1)+3$, with $f(1)=f(-1)=0$ Sure, the homogenous solution can be solved by looking at the ...
3
votes
2answers
950 views

Solving for the closed term solution of a third order recurrence relation with real constant coefficients

How would you solve for the closed term form of $a(n)$ given the general form of the third order linear homogenous recurrence relation with real constant coefficients. ...
2
votes
2answers
395 views

Solving recurrences with boundary conditions

I'm trying to follow CLRS ("Introduction to Algorithms") and I just hit a question in a practice assignment I found online that I just can't make any sense of. Consider this problem: Show that ...
1
vote
4answers
81 views

Solution verification: Prove by induction that $a_1 = \sqrt{2} , a_{n+1} = \sqrt{2 + a_n} $ is increasing and bounded by $2$

I have the following recursive relation (sequence): \begin{align} a_1 = \sqrt{2}, \quad a_{n+1} = \sqrt{2 + a_n} \end{align} My Try: I'm a little skeptical of my manipulations near the end but it ...