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

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2
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1answer
32 views

Find all sequences that satisfy the recurrence relation

Find all sequences that satisfy the recurrence relation $$u_n\cdot (u_{n+1})^2-u_{n+1}-u_n+1=0, \text{with }u_0=1$$ My try First, we find $u_1$, which follows $u_0=1$. $u_0\cdot ...
1
vote
1answer
39 views

How many binary string are there such that there are no k consecutive characters are the same?

Given number $n$ and $k$. Count the number of string with length $n$ such that there are no $k$ consecutive characters are the same. Example with $n = 3, k = 3$, the answer is $6$. ($110, 001, 101, ...
1
vote
1answer
29 views

Solving recurrences for Big-Theta bound

So I am working on my assignment and have gotten stuck. For previous questions I was able to use Master Theorem to get $\Theta$, but can't use the theorem for this question.. I know to get $\Theta$ I ...
1
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2answers
42 views

Recurrence equation of $ T(n) = T(n/2 ) + dn\log_2(n)$

I have the following equation: $$T(n) = T\left({n \over 2}\right) + d n \log_2 n$$ A little investigation: $T(2^1) = 1 + 2d$ $T(2^2) = T(2^1) + 2^2d\times 2 = 1 + 10d$ $T(2^3) = T(2^2) + ...
1
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1answer
64 views

Find $a_{n}$ from a convolution formula

Suppose that $c_{n}$ satisfies the recurrence formula below: $c_2=\alpha$, and $$c_{2n}+c_{2n-2}=\frac{(\alpha)_n}{n!},n\geq2.$$ were $(\alpha)_n = \alpha(\alpha-1)·\cdots·(\alpha-n+1)$ and $\alpha$ ...
6
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1answer
79 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: ...
1
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1answer
62 views

Recursive Solution to Interest with Monthly Deposits

I open an account at a bank with 1% interest compounded monthly. I'm adding $100 to it at the beginning of each month (starting with month 1). (a) Set up a recurrence relation for the amount in the ...
1
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2answers
77 views

Limit of $a_{n+1}=\frac{2a_n^3}{1+a_n^4}$

Let $a_1$ be real, and define $$a_{n+1}=\frac{2a_n^3}{1+a_n^4}$$ How can I prove that this $\{a_n\}$ to have limit. I find it is hard to track. What I can do is just when $a_1=1$ then $a_n=1$; when ...
0
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2answers
49 views

Recurrence Relation $T(n) = T(\frac{n}{2}) + \sqrt{n}$

$T(n) = T(\frac{n}{2}) + \sqrt{n}$ and $T(1) = 1$. Assume $n = 2^k$. $$T(2^k) = T(2^{k-1}) + 2^{k/2}$$ $$T(2^{k-1}) = T(2^{k-1}) + 2^{k/4}$$ ... $$T(2) = T(1) + 2^{k/k} $$ $$T(1) = 1$$ I'm just ...
0
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1answer
38 views

Advanced Recurrence Relations

I have to find a close form of the following Recurrence Relations. $P_{t+1} = P_t - \frac{p}{t}P_t$ With $P_{i+1}=\frac{p}{i}$, for some $i < t$. I tried the unfolding method on Knuth, but it ...
0
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1answer
37 views

consecutive heads

If a coin is tossed 3 times, there are 8 possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT In the above experiment we see 1 sequence has 3 consecutive H, 3 sequences have at least 2 ...
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0answers
13 views

difference equations/inequalities in two variables without constant coefficients

I have a linear inhomogeneous difference inequality with variable coefficients. I was wondering if there are any general methods available for solving it. The case where the inequality is replaced by ...
1
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2answers
50 views

Recurrence Question about $T(n) = T(\frac n2) + nlog(n)$

So I've been working on this recurrence equation and I'm stumped at the end. $T(n) = T(n/2) + n\log(n);\: T(1) = 1;\: n = 2^k$ and log is base $2$. $T(2^k) = T(2^{k-1}) + 2^k\times \log(2^k)$ ...
0
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1answer
30 views

use substitution method to prove an equation is in O(n log2 n)

I am trying to prove that the equation: T(n) = 2T((n/2) +17) + n is O(n log_2(n)) I have to do this by using substitution ...
1
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1answer
27 views

Recurrence Algorithms

What is the best method of solving non standard recurrence algorithms? In particular something like the following: What would be it's tight bound in Theta notation? $$ n \in N\\ T(n) = \sqrt{n} \; ...
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2answers
54 views

How do you solve these recurrence relations for a closed form?

I'm not sure what methods are used to solve recurrence relations for a big-$O$ notation. Thinking about the problem conceptually doesn't really help me, but I feel like I could use some form of ...
0
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1answer
35 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 ...
1
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1answer
22 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 + ...
1
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0answers
102 views

Method for deriving an Exponential Moving Average?

I have a formula for an exponentially weighted moving average function defined recursively as: $S_t = a*Y_t+(1-a)*S_{t-1}$ Where: $a\in (0,1)\cap \mathbb{Q}$ $t$ represents time $Y_t$ is the value ...
0
votes
3answers
116 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
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1answer
25 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
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0answers
37 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 ...
5
votes
2answers
94 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
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1answer
30 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
226 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.
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2answers
31 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
34 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) ...
0
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0answers
16 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 ...
0
votes
1answer
62 views

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

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 ...
1
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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 $g(n)$ ?
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2answers
241 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
19 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
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2answers
26 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
80 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 ...
2
votes
5answers
27 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
42 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
27 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 ...
0
votes
1answer
34 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
3answers
91 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
47 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: $$ ...
2
votes
1answer
93 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, ...
1
vote
1answer
28 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
43 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
56 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 ...
2
votes
1answer
82 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
votes
1answer
53 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
106 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. ...
1
vote
2answers
31 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 ...
0
votes
1answer
25 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 ...