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

learn more… | top users | synonyms

0
votes
0answers
11 views

How do you typically prove recurrence relations?

The median-of-medians algorithm gives a recurrence relation $T(n) = T(n/5)+T(7n/10)+n = O(n)$. If the subgroup was changed to a size 3 or 7, how would this effect the recurrence relation? I came to ...
0
votes
1answer
13 views

Finding a tight upper-bound on $T(n) = 3T(\frac{2}{3}n)$

Can the master theorem be used to prove a tight upper-bound on $T(n) = 3T(\frac{2}{3}n)$? I've drawn the tree for the recurrence and found a sequence: $n + 2n + ...
3
votes
3answers
414 views

Number of Regions in the Plane defined by $n$ Zig-Zag Lines

Fellows of Math.SE, I have been scratching my head at a solution to an exercise in Donald Knuth's Concrete Math. Here is the problem: Here is the solution (I hid it in case someone wants to solve ...
1
vote
0answers
31 views

How do I derive this recursive function?

I have a function $S_t = a*Y_t+(1-a)*S_{t-1}$ Where $a\in (0,1)\cap \mathbb{Q}$, $t$ represents a unit of time $Y_t$ is the value at a time period $t$ I am trying to find $dS_t \over dt$ I have ...
0
votes
3answers
43 views

How would I show that Tn=3^n + 2 is a solution to the recurrence?

Would anyone be able to help me or give me some advice on the following problem: Consider the recurrence with $T_0 = 3$ and $T_{n+1} = 3T_n - 4$ for all $n \in \mathbb{N}$. How would I show that ...
1
vote
1answer
18 views

How can I solve the particular solution of the following recurrence (recursive) relation?

Having $a_n = 3a_{n-1} + 2a_{n-2} + 3·2^{2n-1}$ $a_1 = 12$ $a_0 = 0$ I solved the homogeneous part and got: $a^{{h}}_n = 1/12·2^n - 1/12·1^n$ This is the particular solution that I need to ...
1
vote
0answers
20 views

Using a Recursion Tree to solve the recurrence $T(n) = \sqrt n T(\frac{n}{2}) + 10n$?

I am attempting to solve the above recurence by giving tight $\Theta$ bounds. Assume that the logs here are all base 2! To solve a recursion tree as far as I understand, I need two things. The ...
0
votes
1answer
51 views

How do I say that an infinite-state Markov chain is positive recurrent? [on hold]

I run into this Markov chain while I'm doing my research, and I can't figure out how to find the condition under which this Markov chain is positive recurrent. This is a brief scenario of my ...
0
votes
0answers
13 views

count the permutation which have $k$ maxima

I need some help for the following homework question. A permutation $P (\pi_1\pi_2...\pi_n)$ of {$1,2,...,n$} is given. We say that $j$ is a maxima of $P$ whenever $\pi_j$>$j$. How can I find ...
3
votes
2answers
64 views

partitions and their generating functions and Partitions of n

A partition of an integer, n, is one way of writing n as the sum of positive integers where the order of the addends (terms being added) does not matter. p(n, k) = number of partitions of n with k ...
0
votes
1answer
28 views

How do I solve this recurrence relation

How do I solve the following recurrence relation: T(n)=4T(n-1) - 3T(n-2) I tried using substitution but failed as I was unable to find any "general" i-th term ...
3
votes
3answers
187 views

Closed Form of Recursion

Given that $a_0=2$ and $a_n = \frac{6}{a_{n-1}-1}$, find a closed form for $a_n$. I tried listing out the first few values of $a_n: 2, 6, 6/5, 30, 6/29$, but no pattern came out.
2
votes
1answer
66 views

Recurrence vs Recursive

Say team 1 is studying the recursive characteristics of a function. Team 2 is studying the recurrent characteristics of the same function. Are the 2 teams studying the same thing? I have found for ...
1
vote
1answer
21 views

Solving (for asymptotics) of certain recurrence equations.

I am thinking of examples of the kind where the function occurs multiple times on the R.H.S with different arguments. This is the case where most techniques I know don't seem to work. For example ...
0
votes
1answer
26 views

Recurrence Relation

So I am just making sure I am on the right track with this. I have the recurrence: T(n) = 2T(n-2) + 1 I am trying to solve this recurrence to get the time complexity T(n) = 2(2T(n-4) + 1) + 1 T(n) ...
2
votes
3answers
783 views

Recursively defining the set of bit strings set having more zeros than ones

Question: Recursively define the set of bit strings that have more zeros than ones. I tried it this way: $\Sigma\subset \{0,1\}^*$ Basis step: $0 \in \Sigma$ Recursive step: For any $x\in ...
1
vote
1answer
28 views

$g(n)=\sum_{i=0}^{n-1}g(i)g(n-i-1)$, and $g(0) = 1$, so which is $g(n)$?

I have an equation that: $g(n) = g(0)g(n-1)+g(1)g(n-2) + ... + g(n-2)g(1)+g(n-1)g(0)$ And I also know that $g(0)=1$. How can I derive the close form of function $g(n)$ ?
1
vote
2answers
233 views

Help solving a recursion function T(n) = T(n-2) +3

I have the following recursion function: $T(1) = 0$ $T(n) = T(n-2) + 3$ where n is odd integers I know the closed form of this is: $T(n) = \frac{3n-3}{2}$ but this was purly by guessing. Is it ...
0
votes
1answer
59 views

Gambler's ruin and coin toss

Edit 3. Fixed question to be more clear and include current solution Problem Two players player 1 and player 2 plays a game of fair coin flipping. Player 1 starts with $A$ coins and Player 2 with ...
1
vote
1answer
856 views

Legendre polynomials recurrence relation

How can i get? $$P_{n+1}=xP_n(x)-\frac{1-x^2}{n+1} P'_n(x)$$ $n>=0$ Also know as the leadder equation of the legendre polinomials i tried to use de recurrence relations as: ...
0
votes
1answer
17 views

Number of sequences with n digits, even number of 1's (Continued question)

Some guy asked a very interesting question here before. He was trying to figure out a formula to calculate $a_n$ number of sequences with n digits from $\{1,2,3,4\}$ and an even number of 1's. Which ...
1
vote
2answers
18 views

Recurrence problem with a game of probability [duplicate]

Fair coin flipping (50% on both sides) $P_1$ and $P_2$ plays a few games of fair coin flipping. Assume player $A$ starts with $x$ coins and player $B$ with $y$ coins. Let $P_n$ denote the ...
0
votes
1answer
28 views

recursive sequence - Which approach can I take to solve this equation?

Having this recurrence relation $a_n = 5a_{n-1} - 6a_{n-2} + 4·3^n$ $a_1 = 36$ $a_0 = 0$ How can I solve this? I tried by characteristics roots and got stuck: *making $a_n=r^n$ $r^n = 5r^{n-1} - ...
2
votes
5answers
26 views

Solve recursion with constant added

I have the following problem: Define a sequence $(a_n)$ where $a_1 = 4$ and $a_n = 4a_{n-1} - 4$. Find a closed form for $a_n$. So basically I usually know how to deal with recursions like $a_n = ...
0
votes
1answer
28 views

recursive definition of the relation

Give a recursive definition of the relation greater than on N X N using the successor operators s? I answered this question throw this way: Basis: o ∈ N X N recursive step: if n ∈ N X N, then s(n) ∈ ...
0
votes
1answer
21 views

Prove boundedness of recurrence relation

For a number sequence $\{y_n\}$ we know that $y_{n+1} = 2y_n-y^2_n$ If: $0<y_0<1$ show that $0<y_n<1$ for all integers $n>0$ I've tried solving the recurrence relation, but I couldn't ...
2
votes
1answer
57 views

Solving the recursion $y_n=2ny_{n-1}$ with Wolfram|Alpha

This is blowing my mind away ... this should be easy stuff! Starting with the recursive formula $y_n=2n*y_{n-1}$ where $y_1=5.$ I'm trying to come up with a formula for the series { 5, 20, 120, 960, ...
2
votes
3answers
82 views

Solving a recurrence relation of second order

I have a pattern, which goes: $x_n =2(x_{n-1}-x_{n-2})+x_{n-1}$ and this pattern holds for all $n \ge 2$. I also know that $x_0 = 1 \ and \ x_1 = 5.$ $x_2 = 2(x_1-x_0)+x_1$ $\begin{align} x_3 = ...
0
votes
2answers
42 views

Closed form formula for the given product

I'm working on a recurrence which give me the following solution: $$ f(n)=(2+1)(2+\tfrac12)(2+\tfrac13)\cdots\left(2+\tfrac1{\lg(n)}\right) $$ so for $n=16$, $f(n)$ is just like: $$ ...
1
vote
1answer
304 views

Solving the recurrence $T (n) = \sqrt{n} T(\sqrt{n}) + O (n)$ [duplicate]

I want to show that the requrrence $T (n) = \sqrt{n} T(\sqrt{n}) + O (n)$ is in $O(n \log \log n)$ Here's my attempt: If we expand the recursion tree, at a level $i$, there are $n^{1/2^k}$ ...
1
vote
3answers
2k 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 ...
1
vote
1answer
27 views

Exponential growth with a constant

Some guy opens a bank account with an initial amount of $\$1,000$. Each month he deposits $\$200$ and the bank gives him a monthly interest of $6\%$. I want to find the closed formula. Given this, we ...
2
votes
1answer
64 views

Solving recurrence relation

If I have the following recurrence relation, $$T(n) = T(\lfloor n/2 \rfloor) + T(\lceil n/2 \rceil) + n $$ How would I show that $T(n)\le cn\lg(n)+dn $ for some reals $c$ and $d$?
4
votes
2answers
93 views

How to *really* solve a non-homogeneous recurrence

First let me state that I am not asking about the usual procedure for finding a trial solution to a non-homogeneous recurrence. I have been doing this for many years and can solve all the basic ...
2
votes
1answer
37 views

Unsolveable equation?

If we have the inhomogenous recurrence relation $$f(n+2) - 6f(n+1)+9f(n) = 6*3^{n} + 2^{n} = 2 * 3^{n+1} + 2^n, f(0) = 0, f(1) = 1, n \ge 1$$ Step 1: Find the homogenous solution $f(n) = C_13^n + ...
2
votes
1answer
37 views

Proof of convergence of $a_{n+1} = \dfrac{a_n^2 + 1}{3}$ in $\mathbb{R}$ and finding its limit

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
5
votes
1answer
75 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
0
votes
1answer
21 views

Tight asymptotic upper and lower bounds

I have a equation: $T(n) = 4T(n/3) + n\ln n$ In this equation, I have to give tight asymptotic upper and lower bounds. What does that mean? I know I can apply Master theorem (which gives me theta ...
3
votes
5answers
104 views

Solve $ x_{n+1} - x_n = 2n + 3$

Solve $$ x_{n+1} - x_n = 2n + 3, x_0 = 1, n \ge 0$$ I would try to find a homogen solution and used $$ r^2 - r = 0$$ and got $$x^h_n = A1^n$$ but this seems wrong and I'm stuck on how to continue. ...
5
votes
4answers
88 views

Homework | Find the general solution to the recurrence relation

A question I have been stuck on for quite a while is the following Find the general solution to the recurrence relation $$a_n = ba_{n-1} - b^2a_{n-2}$$ Where $b \gt 0$ is a constant. I don't ...
0
votes
0answers
72 views

Let Cn denote the number of ways of writing a valid list of open and closed parentheses of length 2n

(a) Let Cn denote the number of ways of writing a valid list of open and closed parentheses of length 2n (valid means that at any point along the list, the number of open parentheses must be greater ...
5
votes
4answers
493 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) ...
0
votes
0answers
56 views

Substitution method for solving recurrences

I'm having issues understanding how to solve recurrences via the substitution method. From what I understand, I need to guess the form of the solution first and then use induction to prove that the ...
0
votes
2answers
21 views

What is the intuitive idea behind looking for a solution of the form an=r^n for a linear homogeneous recurrence relation?

In my textbook, under solving linear homogeneous recurrence relations, it says that the basic approach for solving them is to look for a solution of the form an = rn, which yields the characteristic ...
1
vote
2answers
18 views

Recurrence Problem involving multiple dependencies.

I have 3 equations :- $r_n=r_{n-1}+5m_{n-1}$ $m_n = r_{n-1} + 3m_{n-1}$ $p_n = 5m_{n-1}$ The initial values of the sequences are $$r_0=3, m_0=1, p_0=0$$ How can I get the formula to get the nth ...
-2
votes
1answer
85 views

Solving recurrence $T(n) = 3 T (n/3) + n / \lg n$ [closed]

How to solve this relation? I mean the asymptotic solution?
0
votes
1answer
21 views

Solving a recurrence relation with absolute values in it.

The recurrence relation is: Let $\{y_{j}\}_{j\in \mathbb{N}}$ be a sequence of integers and x a real number then define: $P_{1}(x):=y_{1}+(-1)^{1}|x-y_{1}|$ and the general j-step as ...
0
votes
1answer
24 views

limit of $f_n(a) = a^{f_{n-1}(a)}$ as $n$ approaches infinty for small values of $a$

So a friend started, in boredom, calculating values of what I have formalized as $f_n(a) = a^{f_{n-1}(a)}$ (also $f_0(a)=a$) for $a = 1.1$. He noticed that on his calculator it was not changing value ...
0
votes
0answers
12 views

A question on bivariate recurrence

Is there a way to get a closed form of this recurrence (albeit approximate): $$A(n ,k ) = {1 \over {d^k}}A(n-1, k-1) + (1 - {1 \over {d^k}})A(n-1,k)$$ Where, $n,k \ge 0$ and $d \ge 2$ are integers. ...
0
votes
0answers
29 views

Solving Recurrence Relations (Nonlinear?)

I'm not sure the term, but how do you solve a recurrence relation with a multiplicative factor in the index, so as opposed to $a_n=a_{n-1}+a_{n-2}$ we have something like $a_n=a_{\frac{n}{2}}$. I know ...