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

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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
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1answer
47 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 ...
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1answer
35 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 ...
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0answers
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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 ...
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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} = ...
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1answer
33 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 ...
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1answer
35 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 ...
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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 ...
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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 ...
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0answers
39 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 ...
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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
35 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, ...
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2answers
30 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 ...
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0answers
33 views

Solving a non-linear 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 ...
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3answers
97 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 ...
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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 ...
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1answer
57 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. ...
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3answers
47 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 ...
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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 ...
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2answers
43 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 ...
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1answer
53 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 ...
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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) ...
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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 ...
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1answer
34 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 ...
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1answer
32 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: ...
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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) + ...
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1answer
17 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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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 ...
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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 ...
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2answers
81 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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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 ...
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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 ...
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1answer
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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 $a_0=1$ ...
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1answer
40 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 $n>2$, ...
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1answer
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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 ...
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3answers
271 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 ...
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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. ...
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3answers
90 views

Proving the infinite sum of $1/2^i$ without induction

Prove $$\sum_{i=1}^n \frac{i}{2^i} = 2-\frac{n+2}{2^n} $$ Pretty trivial to do with induction, but as a practice problem for solving recurrences we have to do this only by repeating $\sum_{i=1}^n ...
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63 views

Convergence of sequence $x_{n+1}=\sqrt{2x_n+3}$

Let $f:\Bbb R \rightarrow \Bbb R$ be the function $f(x)=\sqrt{2x+3}$. (a) Show that for all $x\in[1,3], f(x)\geq x$. You may use the intermediate value theorem if you like. Let $x_0=1$ ...
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172 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
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1answer
29 views

Quick Recurrences Question [closed]

$$given: T(n)=4T(n/2)+n^2 ;T(1)=1\\=4[4T(n/2^2 )+(n/2)^2 ]+n^2\\=4^2 [4T(n/2^2 )+(n/2)^2 ]+n^2+n^2\\=4^3 [4T(n/2^3 )+(n/2)^2 ]+n^2/4+n^2+n^2\\…\\=4^k T(n/2^k )+$$ Here is where I'm stuck because I'm ...
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Question about Recurrences

$$given: T(n)=T(n-1)+n^3 ; ...
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1answer
110 views

How do I prove that the recurrence contains no perfect square?

Given the recurrence $$a_{n+2} = 14a_{n+1} - a_n - 6$$ with $a_1=1$ and $a_2=8$, how do I prove that none of the $a_n$'s apart from $a_1$ is a perfect square. This is not a homework problem, rather ...
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1answer
118 views

Variation of Tower of Hanoi

I have been reviewing the solution of the following problem for which I have to find a recurrence relation for the number of moves: "In the Tower of Hanoi puzzle, suppose our goal is to transfer all ...
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2answers
40 views

Solving recurrences whose characteristic equations have complex roots

In my Discrete Mathematics lecture notes, there is a section regarding solutions for linear recurrences whose characteristic polynomials have complex roots. There is a particular statement which I am ...
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22 views

function with a recurrence relation

I have this recursive equation: $$\begin{align*} F(m,n)&=F(m,n-1)+F(m-1,n)-F(m-1,n-1-m)\\\\ F(m,0)&=F\left(m,\frac12m(m+1)\right)=1\\ F(m,i)&=0\text{ if }i<0\text{ or ...
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145 views

How can I prove that this recursive sequence converges?

Let $a_0=1$, $a_1=1$ and $a_{n+2}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$. How can I prove that this sequence is convergent? I know that if it's convergent, it converges to $\sqrt{2}$ and I can calculate the ...
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0answers
66 views

Counting all permutations(repeating) with no adjacent elements equal and m majority elements.

Counting all permutations(repeating) of 1 to n which have size n. 1. with no adjacent elements equal, and 2. m majority elements(A element is majority element when it appears max number of times in ...
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845 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...
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2answers
90 views

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

Solve the recurrence $$T(1) = 1, T(2) = 1, T(3) = 1,T(n) = 2T(n-1)+n^2, n > 3$$ I have now, $$T(n) = 2T(n-1)+^2 $$ $$= 2(2T(n-2)+(n-1)^2+n^2$$ $$=4T(n-2)+2(n-1)^2+n^2$$ $$....$$ ...