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

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Rectangle tilling with smaller rectangles

To find the no of ways a rectangle of size 2 $\times $ n can be filled using 1 $\times $ 2 and 2 $\times$ 2 pieces. $$\quad$$ I tried to solve it as a recurrence relation, $a_{2 \times (n+2)} = a_{2 ...
4
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4answers
112 views

Solve the recurrence of the alternating sum $R_n=R_{n-1}+(-1)^{n}(n+1)^{2}$

I have been trying to solve this recurrence for a few hours, but I haven't been able to find the solution yet: $R_0=1$ $R_n=R_{n-1}+(-1)^{n}*(n+1)^{2}$. I have been trying to substitute ...
0
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1answer
24 views

Second order difference equation with a stochastic term

I'm trying to solve a second order difference equation. But there's a stochastic term inside the equation, I was wondering what should the correct way of approaching this problem? Here's the 2nd order ...
1
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1answer
63 views

Asymptotic of an interesting recurrence relation

I want to study the asymptotic behavior of the following recurrence relation: $y_1=1$; $y_{n+1}=y_n+\left(1+\frac{y_n}{n}\right)^{-n}$ for $n\ge 1$. I made an initial attempt and guessed that ...
3
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1answer
48 views

How to show that recurrence $T(n) \in \Omega(n^{0.5})$ using proof by induction?

This is recurrence $T(n)$ $ T(n) = \begin{cases} c, & \text{if $n$ is 1} \\ 2T(\lfloor(n/4)\rfloor) + 16, & \text{if $n$ is > 1} \end{cases}$ This is my attempt to show that $T(n) \in ...
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2answers
25 views

Does this recurrence relation run in $ \Theta(n) $?

This is the recurrence relation I am trying to solve: \begin{align} T(n) & = 2 \cdot T \left( \frac{n}{4} \right) + 16, \\ T(1) & = c. \end{align} I broke this down (i.e., solved this ...
0
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0answers
33 views

How to solve T(n) = 3T(n/4) + c

I found the pattern to this problem being the following... $$3^k T\left(\frac{n}{4^k}\right) + 3^{k-1}c + 3^{k-2}c + \cdots + c$$ I feel like this is wrong but if you can cancel common factors it ...
2
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2answers
41 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
1
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1answer
67 views

Closed form for Numbers in a Triangular Array

I have a particular triangular array $$ \begin{matrix} 1 & \\ 1 & 1 \\ 1 & 2 & 3\\ 1 & 3 & 9 & 15\\ 1 & 4 & 18 & 60 & 105\\ 1 & 5 & 30 & 150 ...
0
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2answers
36 views

Find a recurrence relation and solve it

Let $a_n$ be the nummber of ways that 4 people can throw $n$ eyes together with a die. Every person throw once. Now I want to find a generating function and compute $a_n$ for different $n$. To do ...
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0answers
46 views

How many comparisons are needed for a binary search in a set of 64 elements

Answer: So the recurrence relation for binary search is f(n) = f(n/2) + 2. ...
0
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1answer
53 views

Solving Recurrence Relations with Geometric Series

If given the following problem... $$4T \left(\frac n2\right) + c$$ after getting the pattern down you see the following $$4^k T\left(\frac {n}{2^k}\right) + 3^{k-1}c + 3^{k-2} c + \cdots + 3c + c$$ ...
0
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3answers
45 views

How to solve recurrence relation $a_{k}=7a_{k-1}-10a_{k-2}, \forall k\ge2$ with $a_{0} = a_{1} = 2$

Unfortunately I have no idea where to even start with this. This is my first math class in almost a decade. Can anybody tell me how i would go about solving for the following recurrence relation? ...
1
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1answer
39 views

Directly Obtaining the $n$th Value of a Lucas Sequence

(As an aside: This question lies relatively upon the border between the realms of Computer Science and Mathematics, and thus may be appropriate for StackOverflow as well.) I am in need of a method of ...
2
votes
2answers
52 views

Solve the following recurrence relation:

Solve the following recurrence relation: $f(1) = 1$ and for $n \ge 2$, $$f(n) = n^2f(n − 1) + n(n!)^2$$ How would I go about solving this? Would I need to find a substitution $f(n) =\text{ insert ...
2
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1answer
37 views

Solving a linear nonhomogeneous recurrence relation with troublesome $F(n)$

I am trying to solve the following: $$a_n=5a_{n-1}-6a_{n-2}+2^n+3n$$ The general solution to the homogeneous equation is simple: $$a_n=5a_{n-1}-6a_{n-2} \rightarrow \\ r^2-5r+6=0 \rightarrow ...
2
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0answers
21 views

Determining the sequence that yields a balanced search tree in the form of a recurrence / sequence

Let's say I have a sequence of (distinct) monotonically increasing numbers S. I'll want to add them sequentially to a Binary Search Tree (BST) but as the numbers ...
0
votes
1answer
16 views

Recurrence Relations with Geometric Series

if we have a situation where something is like this $2^k + c(2^{k-1} + 2^{k-2} + 2^{k-3} + ... + 1)$ since in this case $r > 1$ then in Computer Science we look at $\sum_{i=1}^{n} r^{i} = ...
1
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1answer
35 views

Solving recurrence relations with two variables

whenever I've had to solve recurrence relations, I've kind of just messed around with it until it works. I have a more complicated case, and I was wondering if there are general strategies someone ...
1
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1answer
38 views

Simplified summation formula.

Suppose I have the following recursive formula: $$A(n)=-\sum_{k=0}^{n-1}\binom{n}{k}\frac{1+(-1)^{n-k}+2(n-k)(-1)^{n-k-1}}{2}A(k)$$ Then I can combine the negatives to get ...
4
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0answers
71 views

What is the explicit formula (solution) to this recursively defined binary matrix?

My question concerns the following binary matrix (call it matrix $A$). Or rather the entire family of such matrices, for some number of columns $n$ and rows $2^n$. The ellipses indicate that the ...
1
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4answers
85 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 ...
2
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0answers
43 views

Generating function for recurrence in two variables

Given characteristic polynomial for the recurrence in two variables (say $F(x,y)$) $$ (y^2-1)^x $$ and initial values can generating function for $F(x,y)$ be derived? I know how to do it for a ...
0
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1answer
52 views

Solve this recurrence relation

Solve the following recursions: $a_{n+1}=3a_n-a_{n-1}-1$ and $a_{n+1}=4a_n-a_{n-1}-1$. (These are to be solved separately, not simultaneously) I tried using generating functions but it got messy. Any ...
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1answer
39 views

How to prove the characteristic equation based solution of recurrence relations?

What is the proof for / where might I find the proof to: Let $c_1, c_2,..., c_k$ be real numbers. Suppose that the characteristic equation $$r^k-c_1 r^{k-1}-...-c_k=0$$ has $k$ distinct roots $r_1, ...
0
votes
2answers
31 views

Prove T(n)= T(n-2)+k is O(n) for all n >1

I'm stuck on trying to prove that $ T(n)= T(n-2)+k$ is bounded by $O(n)$ for all $n >1$ I expanded it out to reach the following guess: $T(n) = ((n-2)/2)k $ though when I try to prove ...
0
votes
1answer
45 views

Solving a non-homogeneous recurrence relation

I have the following non-linear homogeneous recurrence relation: $a_n = 6a_{n-1} - 12a_{n-2} + 8a_{n-3} + 2^n$ And I need to solve it by giving a general form . So I get the process. First I solve ...
2
votes
3answers
101 views

Use the generating function to solve a recurrence relation

We have the recurrence relation $\displaystyle a_n = a_{n-1} + 2(n-1)$ for $n \geq 2$, with $a_1 = 2$. Now I have to show that $\displaystyle a_n = n^2 - n +2$, with $n \geq 1$ using the generating ...
0
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1answer
29 views

For a solution of linear recurrence relation, $\lim_{n\to\infty}a_n^{1/n}$ is a zero of a related polynomial

On page 134 of J.H. van Lint's book A Course in Combinatorics, it says from $a_n=5a_{n-1}-7a_{n-2}+4a_{n-3}$ $(n\ge5)$, we find that $\lim_{n\to\infty}a_n^{1/n}=\theta$, where $\theta$ is the ...
1
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1answer
58 views

solving a recurrence without initial conditions

I have been working on this problem for two days... I can only get as the characteristic part of the recurrence, I just can't figure out a proper guess for the particular solution. ...
0
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3answers
56 views

Recurrence relation - equal roots of characteristic equation

I have the following problem: Solve the following recurrence relation $f(0)=3$ $f(1)=12 $ $f(n)=6f(n-1)-9f(n-2)$ We know this is a homogeneous 2nd order relation so we write the ...
0
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1answer
52 views

non-homogenous linear recurrence relation general questions

what happens if you have both repeated and non-repeated roots? i know there are different forms for both, so if given roots say $5, -3, -3, -3$ would it then be $A(5)^n + B(-3)^n + Cn(-3)^n + Dn^2 ...
1
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2answers
44 views

Combinatorics Recurrence relation

Let $h_n$ be a number sequence where $h_n = 3h_{n-1} - 2h_{n-2}$ with $h_0 = 0$ and $h_1 = 1$. Compute the ordinary generating function of $h_n$ and then using the generating function compute a ...
2
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1answer
58 views

Find generating functions for the Perrin and Padovan sequences

The Perrin sequence is defined by $a_0 = 3, a_1 = 0, a_2 = 2$ and $a_k = a_{k-2}+a_{k-3}$ for $k \ge 3$. The Padovan sequence is defined by $b_0 = 0, b_1=1, b_2=1$ and $b_k=b_{k-2}+b_{k-3}$ for ...
1
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1answer
89 views

evaluate trigonometric

To all the genius out there , here is a question about expresssing summation of hyperboilc functions : First of all, I've already proved that: $$\sinh(x + 1)- \sinh(x) = (-􀀀1 + \cosh (1)) \sinh(x) ...
6
votes
0answers
100 views

An Impossible Sequence of Prime Powers

Let $x_1,x_2,\ldots$ be a sequence of positive integers that satisfies the recurrence relation $$x_{n+1}=2x_n(x_n-1)+1$$ for all positive integers $n$. It seems impossible that every term in this ...
1
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1answer
35 views

Tools for solving recurrent expresions

I've got a problem involving a recurrent expression. I would like to find a solution of $x_t$ that let me take derivatives or finding the minimum of the function. Does anybody know tools for solving ...
0
votes
1answer
35 views

Recurrance relation with non-integer part

Apologies if this has already been asked, but I'm stuck. I have a recurrance relation that I want resolve to the form $a_n=f(n,\gamma)a_0$ but I don't know what to do. The relation is: ...
1
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1answer
25 views

Writing a tight bound for a recurrence relation

$$\begin{align}T(n) &= 2 \cdot T(n-1) + 1\\ &= 2^2\cdot T(n-2)+2+1\\ &= 2^3\cdot T(n-3)+2^2+2+1\\ &= 2^4\cdot T(n-4)+2^3+2^2+2^1+2^0\end{align}$$ general form: $2^n\cdot T(0) + ...
0
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1answer
18 views

Recurrence Relation with Variable Coefficient Help

I'm sure that this question is very simple, but there are no example like it in the course material and I'm not really sure what I'm looking for online. $x_n=2^n x_{n-1}, x_0=3$ If anybody could ...
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1answer
51 views

Edited-How can I solve polynomial recurrences like $f(n+1)=\frac{2f(n)}{f(n)+1}$

Can anybody tell me the systematic way of solving this recurrence. $$f(n+1)=\frac{2f(n)}{f(n)+1}$$ I looked over the internet, but could not find the answer. Thanks {Edit- I am sorry, previously I ...
0
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1answer
51 views

Mini Tetris Winning Configuration

So here's the problem: A winning configuration in the game of Mini-Tetris is a complete tiling of a 2 x n board using only the three shapes shown in Figure 1. By allowing rotations, there can be ...
1
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2answers
88 views

Recursively defining sets of strings discrete math

So here are the two problems: Recursively define the set of bit strings K that do not have 00 as its substring. How many bit strings of length 10 are included in the above set K? Can someone ...
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1answer
55 views

Binary strings and recurrence relations [closed]

So the problem is: How many binary strings of length n contains 111? Give a recurrence relation Tn, where Tn is the number of binary strings of length n that contains 111. How could we possibly ...
1
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1answer
41 views

Are these recursive sequences convergent?

Fix an integer $k > 1$. Suppose $a_1,\ldots,a_k > 0$ and for $n > k$ we define $$a_n = 1/a_{n-1} + 1/a_{n-2} + \ldots + 1/a_{n-k}$$ Are these recursive sequences always convergent for any ...
1
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1answer
26 views

Recurrence Relation and finding cosine of a function of them.

What if we are given $$a_{r+1}=\sqrt{\frac12(a_r+1)},r\in\{0\}\cup\mathbb N$$ How to find: $$\chi=\cos\left(\frac{\sqrt{1-a_0^2}}{\displaystyle\prod_{k=1}^{\infty}a_k}\right)$$ My try, let $a_0=1$ ...
1
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1answer
41 views

Why is Mergesort $O(n)$ rather than $O(n\log{n})$?

Assume we want a divide-and-conquer algorithm that finds the max and min of a set $S$ with $n = 2^k$ elements, e.g. mergesort. The recurrence for time complexity is $T(n)=2*T(n/2) +2$, for $n>2$, ...
0
votes
1answer
23 views

Prove that $w/w_0$ (no idle over minimum possible) $\le 2-1/n$ for any set of tasks on an n processor system

$w/w_0 $ $\le 2-1/n$ I've noticed this problem in a couple of discrete math and algorithm analysis textbooks. Many of them prove it for n=2, but I want to prove it for all n. The idea is that we ...
1
vote
3answers
286 views

Solving this recursive relation

I want to solve this recursive relation: $$i_{n+1}=4i_{n}+9$$ where the $i_1=t$ that $t \in \mathbb{N}$ I tried to make like relation about Tower of Hanoi, but no good thing happened. How can I do ...
2
votes
2answers
64 views

Solving the recursion $T(n) = T(n-1)\cdot T(n-2)$

Given $T(1) = a$ and $T(2) = b$, solve for $T(n)= T(n-1)\cdot T(n-2)$ [For the sake of clarity,that is $T(n-1)$ multiplied by $T(n-2)$ ] It was asked in one of the entrance tests for a PHD program. ...