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

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3answers
26 views

Where is the error in finding the particular solution to this recurrence relation?

The question is to write the general solution for this recurrence relation: $y_{k+2} - 4y_{k+1} + 3y_{k} = -4k$. I first solved the homogeneous equation $y_{k+2} - 4y_{k+1} + 3y_{k} = 0$, by writing ...
5
votes
5answers
644 views

Solving a set of recurrence relations

I have the 7 following reccurence relations: $A_n = B_{n-1} + C_{n-1}$ $B_n = A_n + C_{n-1}$ $C_n = B_n + C_{n-1}$ $D_n = E_{n-1} + G_{n-1}$ $E_n = D_n + F_{n-1}$ $F_n = G_n + C_n$ $G_n = E_n + ...
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2answers
45 views

Use Bonnet recursion formula to prove by induction

Use Bonnet recursion formula: $P_{n+1}(x) = \frac{2n+1}{n+1} x P_n(x) - \frac{n}{n+1} P_{n-1}(x)$ to prove by induction 1) $P_n(1) = 1$ for all $n$ 2) $P_n(-x) = (-1)^n P_n(x)$ for all $n$ an for ...
1
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1answer
27 views

Hypergeometric function relation knowing initial value?

Is there a relationship or recurrence relation I can use to solve for $$\, _2F_1(b,r+k;a+b+k;p)$$ as a function of $k$, with known value of when $k=0$ $$ \, _2F_1(b,r;a+b;p) = f_0$$ (a,b,r,p) are ...
0
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2answers
54 views

Using generation functions solve the following difference equation

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

Could someone explain me this induction.

I'm trying to understand a paper called "Diameter of Polyhedra: Limits of Abstraction" available here : http://sma.epfl.ch/~eisenbra/Publications/designs.pdf My problem is with the first two ...
3
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4answers
106 views

Closed form of a recurrence relation using generating functions

It's been awhile since I have done this. The sequence is $\displaystyle a_n = a_{n-1} + 5~a_{n-2}$ with $a_{0}=0$ and $a_{1}=1$. I found the generating function to be $\displaystyle G(x) = ...
0
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3answers
464 views

Finding Particular Solutions to Non-Homogeneous Recurrence Relations

Could anyone assist me in solving the following recurrence relations? $a_n = 3a_{n-1} - 2a_{n-2} + 2^n n^2$ $b_n = -nb_{n-1} + n!$ Specifically, I am not sure how to find the particular solutions ...
0
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3answers
553 views

non homogeneous recurrence relation

I am trying to solve the non-homogeneous linear recurrence relation: $$f(n) = 6f(n-1) - 5,\quad f(0) = 2.$$ How do I go about doing it? This is so different from solving a homogeneous recurrence ...
5
votes
4answers
163 views

Finding the general term of two related recurrence relations

I'm trying to find the general term of the recurrence relations $\quad a_{n+1}=a_n+\text hb_n$ $\quad b_{n+1}=b_n-\text ha_n $ $\quad a_0=0, \quad b_0=1$ I tried finding the terms, ...
9
votes
3answers
2k views

Solving recurrence relation in 2 variables

We already know how to solve a homogeneous recurrence relation in one variable using characteristic equation. Does a similar technique exists for solving a homogeneous recurrence relation in 2 ...
1
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2answers
14 views

Finding a particular solution for non-homogeneous recurrence relation [on hold]

The recurrence relation that I have is $$T(n) -5\ T(n-1) + 6\ T(n-2) = 2n.$$
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0answers
25 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: ...
0
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2answers
32 views

2nd order homogeneous difference equation

$$a_{n+2} = 9a_{n+1} - 18a_n,\quad n\geq 0,\,\,a_0=1,\,\, a_1=3$$ I got to the point where i moved all to LHS which gives me $a_{n+2} - 9 a_{n+1} + 18 a_n$ (correct me if I'm wrong). I then ...
2
votes
2answers
213 views

Solve the following non-homogeneous recurrence relation:

Find the solution to the following non-homogenous recurrence relation: $a_{n+2} - 4a_{n+1} + 4a_{n} = 2^n$ for $a_0=1, a_1 = 2$. I have found from the characteristic polynomial the general homogenous ...
2
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3answers
632 views

Summation in a recurrence relation

edited to reflect advice from the comments: While working on a generalization of a tiling problem, I generated a recurrence relation to describe the total number of possible tilings. The relation ...
0
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2answers
33 views

Higher order recurrence relation

I have the following non-homogenous recurrence relation and I'm trying to solve it using characteristics roots method : $a_n = 10a_{n-1} -37a_{n-2} + 60a_{n-3} -36a_{n-4} +4$ for $n \ge4$ and $ a_3 = ...
-2
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1answer
22 views

Which is a linear and homogeneous recurrence?

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

difference equation( recurrence relation)

Let $y_n$ satisfy the nonlinear difference equation: $$(n+1)y_n=(2n)y_{n-1}+n.$$ Let $u_n=(n+1) y_n$. Show that $$u_n= 2u_{n-1}+n.$$ Solve the linear difference equation for $u_n$. Hence find ...
1
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2answers
58 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: $$ v_n = v_{n-1} + v_{n-2} + v_{n-3} $$ ...
4
votes
3answers
314 views

How do I resolve a recurrence relation when the characteristic equation has fewer roots than terms?

I know how to solve "simple" recurrence relations. For instance, say you have: $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$ We can write the characteristic equation as: $$3x^{n-1} - ...
3
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0answers
135 views

General overview about Recursion (free online texts)

I'm looking for free online texts about recursion. What I'm looking for formal definitions* of "all" (most of) the different types of recursion and from different points of views like Category ...
4
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3answers
804 views

Recurrence relation (linear, second-order, constant coefficients)

Q1. Find the general solution to the difference equation $$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$ Q2. Solve the difference equation $$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$ I am ...
2
votes
2answers
59 views

recurrence relation Homework question 1

This is a HW question Find the recurrent relations for $a_n, n\geq 0$ where $a_n$ is the number of $n$character upper case words that contain exactly one $A$ We are only required to find the ...
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0answers
12 views

Series expansion in a recurrence relation (Lines in a plane)

L The recurrence is therefore L0 = 1 ; Ln = Ln−1 + n , for n > 0. The known values of L1 , L2 , and L3 check perfectly here, so we'll buy this. Now we need a closed-form solution. We could play ...
2
votes
2answers
23 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 ...
1
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3answers
90 views

Recurrence relations (Big-O notation)

Say I'm given a recursive function such as: function(n) { if (n <= 1) return; int i; for(i = 0; i < n; i++) { function(0.8n) } } ...
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0answers
23 views

Approximating the function $ f(x) = \frac{1}{1+a^2x^2} \text{with } a=4 \text{ in the interval }[-1,1]$ with Legendre Polynomials

Given: $$ f(x) = \frac{1}{1+a^2x^2} \text{with } a=4 \text{ in the interval }[-1,1]$$ Approximate the function $f(x)$ in the least squares sense using legendre polynomials up to order 2. The ...
1
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0answers
13 views

Almost complete multivariate recurrence solution…

$$ \gamma c_{jm_1m_2} = s^+_{jm_1}c_{j(m_1+1)m_2} - s^+_{jm_2}c_{jm_1(m_2+1)}\\ \gamma d_{jm_1m_2} = s^-_{jm_1}d_{j(m_1-1)m_2} - s^-_{jm_2}d_{jm_1(m_2-1)} $$ where $s_{jm_i}^{\pm} = ...
0
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1answer
25 views

help me finding the limit of sequence question

A 1= 1 and {A n}=root[1 + {A n-1}] B 1= 2 and {B n}=root[2*{B n-1}] help me Since I am studying math recently I need person`s help
2
votes
3answers
59 views

solution of a recurrence

How might one solve the recurrence $x_{n+1} + x_n + 2^n = 0$ given the necessary initial conditions ($x_0$)? Possible ideas I have in mind: 1) Generating functions 2) Discrete Laplace ...
1
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1answer
20 views

Does the order I multiply the characteristic equation's factors in the homogeneous solution matter?

I've been doing a recurrence relation exercise in my book. Doing some steps and comparing them to the ones taken by the book. $$T(0) = 1$$ $$T(1) = 2$$ $$T(k) - 7T(k-1)+10T(k-2)=6+8k$$ ...
2
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0answers
32 views

Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
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5answers
279 views

What particular solution should I guess for this relation?

Just trying to solve a non-homogeneous recurrence relation: $$f(n) = 2f(n-1) + n2^n$$ $$f(0) = 3$$ Characteristic equation is: $$f(n) - 2f(n-1) = 0$$ $$a-2 = 0$$ $$a = 2$$ Homogeneous ...
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0answers
16 views

order definition [on hold]

i'm in desperate need of your much appreciated expertise with trying to define the following order relation, guess it might be somewhat near a lexicographical order, not a straightforward alphabetical ...
0
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2answers
40 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 ...
0
votes
2answers
45 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 ...
3
votes
2answers
80 views

Deriving a (tricky, I think?) recurrence relation

I'm having trouble trying to derive a recurrence relation for a problem I'm looking at. "Let $h_n$ be the number of ways of packing a bag with $n$ fruits (either apples, oranges, bananas, or pears), ...
1
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1answer
37 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 ...
1
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1answer
13 views

About the particular solution given an homogeneous solution in a recurrence relation.

If your recurrence relation's characteristic equation factorizes to $$(x+1)(x-5)^3 = 0$$ and $h(n) = 3+2n \implies f_p(n) = d_0+d_1n$ $h(n) = 7n+3^n \implies f_p(n) = ...
1
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1answer
16 views

How to solve non-homogeneous recurrence relations?

I have been looking around for a general method to solve non-homogeneous recurrence relations. Solving non homogeneous recurrence relation seems to be having almost the same problem as me. There is ...
0
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0answers
20 views

Non homogenous Recurrence Relation Problem

Consider the recurrence relation $$b[n] = b\left[ \frac{n}{2} \right] + b \left[ \frac{n+1}{2} \right] + 2$$ for $n > 1$ with $b[1] = 0$. Solve the recurrence in the case that $n$ is a power of $2$ ...
0
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1answer
20 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 ...
1
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1answer
25 views

Need help with recurrence relations in general as well as specific problem

I am supposed to find the unique solution for a recurrence relation and I literally have no idea what to do. Here is what the professor did for us in class: \begin{align*} 3a_{n+1} - 4a_n &= 0 ...
2
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5answers
185 views

Deriving Closed Form for a Recursion via Generating Functions

Consider (1) $a_{n+2} = 2a_{n+1} - a_n + 4n3^n$ with $a_0 = a_1 = 1$. Using generating functions and setting $A(x) = \sum a_nx^n$ we obtain $$\begin{align*}&\quad\sum a_{n+2}x^{n+2} = ...
2
votes
2answers
68 views

Finding the coefficient in the closed form of the generating function

I try to solve the recursion $a_n=5a_{n-1}+5^n$ with $a_0=1$ with generating function, but I could not find the coefficient of $x^n$ in the closed form \begin{eqnarray*} ...
2
votes
3answers
115 views

Problem with generating functions and binary recurrences

I am considering the following recurrence: $a_0 = 1$; $a_1 = 2$ $a_{n} = 2 (a_{n - 1} + a_{n - 2})$ Then I proceeded with the generating function: $F(x) = \displaystyle\sum_{n = 0}^\infty a_n x^n ...
0
votes
0answers
36 views

Total Time for a ball that bounces from a height of 8 feet and rebounds to a height 5/8 [duplicate]

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
2
votes
3answers
117 views

Functional equations and generating functions

The problem asks to find the functional equations for the generating functions whose coefficients satisfy $$ a_n = \sum_{i=0}^{n-1} a_i a_{n-1-i}\,\, (n\geq1), a_0 = 1 $$ There's an example that's ...
2
votes
1answer
83 views

How find this function $f(2^m)=?$

show that:there exists unique function $f:N^{+}\to N^{+}$,such $$f(1)=f(2)=1$$ and $$f(n)=f(f(n-1))+f(n-f(n-1))),n=3,4,\cdots$$ and Find $f(2^m),m>2,m\in N$ My try: let $n=3$ ,then we have ...