1
vote
2answers
17 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
58 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 ...
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
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 = ...
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
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 + ...
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 ...
0
votes
1answer
22 views

How many partitions are there?

How many partitions are there for $\{1,\cdots,100\}$ for $3$ sets, $A,B,C$, such that $A$ cannot contain consecutive numbers ($\left|a-b\right|=1$) Anyway, I thought about using recurrence ...
0
votes
1answer
79 views

Prove this recurrence relation? (catalan numbers)

$$C_0 = 1,\quad C_{n+1} = C_0C_n + C_1C_{n−1}+ \cdots + C_kC_{n−k} + \cdots + C_nC_0\text{ ?}$$ Where Cn denotes the number of ways of writing a valid list of open and closed parentheses of length ...
1
vote
1answer
42 views

Segner's Recurrence Relation [closed]

Why is Segner's Recurrence Relation formula valid. Does anyone know how to prove it? I can't seem to understand why this formula works/is true. $$C_0 = 1,\quad C_{n+1} = C_0C_n + C_1C_{n−1}+ \cdots ...
2
votes
1answer
28 views

Time Complexity of one Example Code

i see an example on my note for calculating Time Complexity, but i couldn't understand. anyone could help me.
4
votes
2answers
29 views

Solve $T(n) = 1 +\sum_{i=0}^{n-1}T(i)$

For the recurrence defined by $$T(n) = 1 +\sum_{i=0}^{n-1}T(i)$$ Apparently $T(n) = 2^n$ .. but I cannot see it. This recurrence pops up during analysis of the Rod Cutting Problem. I keep looking to ...
1
vote
1answer
44 views

Find all solutions of the recurrence relation $a_n = 5a_{n-1} - 6a_{n-2} + 2^n + 3n$

Question: Find all solutions of the recurrence relation $a_n = 5a_{n-1} - 6a_{n-2} + 2^n + 3n$ (Hint: Look for a particular solution of the form $qn2^n + p_1n + p_2$, where $q, p_1, p_2$ are ...
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
0answers
35 views

Concrete math generalized josephus recursion understanding 1.15

I am studying through the josephus problem in concrete math , Here is the equation of binary form $$f(1) = α ;$$ $$f(2n + j) = 2f(n) + β_j ,$$ $$\text{ for } j = 0, 1 \text{ and } n \geq 1$$ this ...
4
votes
0answers
51 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
0
votes
1answer
67 views

Solving this recurrence relation

please tell me a way to solve this recurrence $$n\cdot R_n=C_1\cdot R_{n-1}+C_2\cdot R_{n-2}.$$ $C_1$ and $C_2$ are constants.. There is an $n$ there in the left hand side.. it makes mess. I tried ...
0
votes
1answer
28 views

Change of variables in function $T(n)$.

I've been given this recurrence to solve: $T(n) = T(\sqrt n) + \theta(lglgn)$ And I'm told that the way to solve it is to let $m = lgn$, so that the recurrence can be rewritten as follows: $S(m) = ...
0
votes
2answers
28 views

Find the values of $r_1, r_2, r_3$ in the recurrence relation $a_n = r_1 a_{n-1} + r_2 a_{n-2} + r_3 a_{n-3} $

Consider the recurrence relation $a_n = > r_1 a_{n-1} + r_2 a_{n-2} + r_3 a_{n-3} $ for $n \ge 3$. The roots of the characteristic polynomial are $x = -1$ of multiplicity 2, $x= 3$ of ...
1
vote
1answer
22 views

Cardinality of a set with a recurrence relation.

Let $A = \left\{ f\in \mathbb{N}\rightarrow \mathbb{C} \mid \forall n\in \mathbb{N}. f(n+3) + 3f(n+1) = f(n+2)+f(n) \right\}$ What is $\left|A\right|$? Well, I tried to treat $f$ as a recurrence ...
3
votes
2answers
57 views

Solving combinatorical problem using characteristic polynomial

How many $6$ length strings above $\left\{1,2,3,4\right\}$ are there such that $24$ and $42$ aren't allowed. The suitable recurrence relation for this problem is: $a_{n+2} = 2a_{n-1} + ...
0
votes
2answers
138 views

Recurrence Relation Solving Problem

Can anyone help me in solving this complex recurrence in detail? $T(n)=n + \sum\limits_{k-1}^n [T(n-k)+T(k)] $ $T(1) = 1$. We want to calculate order of T. I'm confused by using recursion tree ...
0
votes
2answers
42 views

Divide and Conquer Recurrence Relation help?

So the divide and concur recurrence has to be of the form H($n$) = $a$H($n$/$b$) + $cn^d$. I already figured out that $a$ = 4 and $b$ = 2. I am really stuck on how to find $cn^d$ however. I ...
0
votes
1answer
28 views

Particular solution of recurrence relation

I've got this recurrence relation: $$M_n = M_{n-1} + n(2n-1)|M_0 = 0$$ and can't think of any form of particular solution to get a solvable constant. With $M_n^H = K$being the homogeneous part of ...
10
votes
3answers
178 views

Show that $2^n(\cos^n(\frac{2\pi}{9})+\cos^n(\frac{4\pi}{9})+\cos^n(\frac{8\pi}{9}))\in\mathbb{Z}$

$a_n=2^n\left[\cos^n\left(\dfrac{2\pi}{9}\right)+\cos^n\left(\dfrac{4\pi}{9}\right)+\cos^n\left(\dfrac{8\pi}{9}\right)\right]$. Show that $a_n\in\mathbb{Z}$ for all $n\in\mathbb{Z}$. Find the last ...
1
vote
1answer
36 views

Help with recurrence equation

I need to solve the following recurrence equation $p_i =\begin{cases} r(p_{i-1}+p_{i+2}) &\mbox{if } i \text{ is odd} \\ (1-r)(p_{i-1}+p_{i}) & \mbox{if } i \text{ is even} \end{cases} i ...
1
vote
2answers
25 views

Not sure how to do Non-Homogeneous Recurrence Relations

I have a sample exam paper, and the answer is given, but I can't work out the answer from the question: Find the solution of: $a_n = \frac{1}{3}a_{n-1} + 2$ using $a_0 = 4$ Given Answer: $a_n = ...
1
vote
1answer
39 views

Recurrence Relation all general solutions

I need some help solving the following recurrence relation: $a_n = 4a_{n-1} - 4a_{n-2} + (n+1)*2^n$ What I've tried: a) Find the general solution of the associated linear homogenous recurrence ...
1
vote
2answers
138 views

Find an explicit formula for the recursive sequence (tips?)

Problem: A sequence is defined recursively as follows: Sk = 2k - Sk - 1, for all integers k greater than or equal to 1 S0 = 1 Use iteration to guess the explicit formula for the sequence. Use ...
1
vote
1answer
74 views

Need help with recurrence relation

So the question is: "Find a recurrence relation for the number of ways to pick $n$ objects from $k$ types with at most 3 of any one type" I think I figured out what it would be excluding the last ...
0
votes
1answer
120 views

find the recurrence relation (homework)

I'm new to recurrence relations and I'm having trouble figuring out this problem: Find a recurrence relation for the number of ways to make a stack of green, yellow, and orange napkins so that no two ...
0
votes
2answers
28 views

Find $f(n)$ when $n = 2^k$ where $f$ satisfies the recurrence relation $f(n) = f\left(\frac{n}{2}\right) + 1$ with $f(1) = 1$

Given: $f(1) = 1$. Answer: $$f(2) = f(1) + 1 = 1 + 1$$ $$\ldots$$ $$f(4) = f(2) + 1 = 1 + 1 + 1.$$ How do I find the value of $f(n)$ where $n$ is an odd integer? Let say $f(3) = ...
2
votes
2answers
75 views

Finding the recurrence relation for number of ways to deposit n dollars

Question: A vending machine dispensing books of stamps accepts only one dollar coins, 1 dollar bills and 5 dollar bills. a) Find a recurrence relation for the number of ways to deposit n dollars in ...
0
votes
1answer
10 views

How does one find annihilators for recurrence relations?

I've recently taken an interest in the Method of Annihilators for solving recurrence relations. However, I taught myself using nearly solely this Power Point presentation. Thus, my knowledge is ...
1
vote
4answers
47 views

Help understanding Recursive algorithm question

We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$ For $n\geq0$, let $c(n)$ be the total number of additions for calculating $f(n)$ ...
0
votes
1answer
77 views

Induction Proof of: Find $f(n)$ when $n=2^k$, where $f$ satisfies the recurrence relation $f(n)=f(\frac{n}2)+1$ with $f(1)=1$

How can I proof this using regular induction and strong induction? $$f(2^k)=f(\frac{2^k}2)+1=f(2^{k-1})+1$$ $$f(2^{1-1})=f(2^0)=f(1)=1$$ $$f(2^{2-1})=f(2^1)=f(2)=f(1)+1=2$$ ...
3
votes
4answers
172 views

Solving the non-homogeneous recurrence relation: $g_{n} = 12g_{n-2}-16g_{n-3}+6\cdot 2^n+25n$

$g_{n} = 12g_{n-2}-16g_{n-3}+6\cdot 2^n+25n$ With initial conditions $g_{0} = 23, g_{1} = 37, g_{2} = 42 $ This is a practice question I'm working on, and I'm running into absurd amounts of ...
2
votes
5answers
682 views

Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
2
votes
1answer
57 views

Find the generating function for a series , given a recurrence relation

I am solving a problem on an Online Judge. The problems solution boils down to find the solutions to the following recurrence relation: ...
-1
votes
1answer
34 views

Which is a linear and homogeneous recurrence?

Which of the following choices is a linear and homogenous recurrence? $1)$ $A_n = A_{n-1} + 4A_{n-2} + 3n$ $2)$ $A_n = n + 1$ $3)$ $A_n = (A_{n-1})^2$ $4)$ $A_n = 5A_{n-1} + A_{n-2} + 3A_{n-3}$
-2
votes
2answers
115 views

Using generation functions solve the following difference equation

Using generation functions solve the following difference equation $$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
1
vote
2answers
84 views

Get the Nth term of a sequence 1,2,4,7,13,24…

I have a sequence: 1,2,4,7,13,24,44,81, ... and I think it's like a Fibonacci sequence, however you add three number together and not two ("Tribonacci"?). So: $$ v_n = v_{n-1} + v_{n-2} + v_{n-3} $$ ...
2
votes
2answers
53 views

Recurrence relation practice problem that I can't figure out

Thanks for taking the time to look at this problem. I'm trying to prepare for a test on Monday by doing some extra odd numbered problems from my textbook. I'm having a lot of trouble trying to solve ...
0
votes
2answers
49 views

Recurrence Relationship Questions

Consider the recurrence defined by: $$G_0 = 0\\ G_n = G_{n-1} + 2n - 1$$ Determine what Gn is for several values of n to determine a formula for Gn. $2n$ $n$ $2n-1$ $n^2$ *I believe this one is ...
1
vote
1answer
45 views

Simple Recurrence Questions [closed]

$r$ is a real number, define a recurrence relationship for $A_n$ $$A_0 = 1\\ A_n = r\cdot A_{n-1}$$ Question: What is the value of $A_4$ $4(A{n-1})$ $r^4$ $1$ $4r$ I've pretty much eliminated ...
0
votes
1answer
22 views

Find a sequence $a$ so that $a_n = s \Delta a_n $.

Let $s$ be a real number $ s \ne 0 $. Find a sequence $a$ so that $a_n = s \Delta a_n $ and $a_0 = 1$. Any help with this question will be great. This is my first time doing recurrence relations ...
0
votes
2answers
46 views

Two sequences $a$ and $b$ for which $\Delta a_n = \Delta b _n$

Find two different sequences $a$ and $b$ for which $\Delta a_n = \Delta b_n$ for all of $n$. This is my first time doing recurrence relations, so if anyone could provide some thorough and clear ...
0
votes
2answers
42 views

Solving a Linear Recurrence Relation

I made quick progress on this, and then of course got stumped, so here's the problem: $$a_0 = -1, a_1 = -2, a_n = 4a_{n-1} - 3a_{n-2}$$ So, following the way I was taught to solve this type of ...
3
votes
2answers
480 views

Linear Homogeneous Recurrence Relations and Inhomogenous Recurrence Relations

I'm having some difficulty understanding 'Linear Homogeneous Recurrence Relations' and 'Inhomogeneous Recurrence Relations', the notes that we've been given in our discrete mathematics class seem to ...
0
votes
2answers
34 views

How to get the characteristic equation from a recurrence relation of this form?

I've been getting the characteristic equation from relations of the form $$U_n=3U_{n-1}-U_{n-3}$$ Thanks to this question I made before: How to get the characteristic equation? Now, I recently ...