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

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1answer
94 views

Solve the recurrence relation $x_{n+2} -3x_{n+1} + 2 x_n = n$

Solve $$x_{n+2} -3x_{n+1} + 2 x_n = n$$ when $x_0 = 1$ and $x_1 = 0$. I started with the homogen solution: $$r^2 -3r +2 = 0$$ So $$x_n^h = A1^n + B2^n$$ I know that $x_n = x_n^p + x_n^h$ But I ...
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1answer
39 views

Recursion and divisibility by $2^n$

A team plays a series of games, each of which results in either a win (W), a draw (D), or a loss (L). Let $S_n$ denote the number of possible sequences for a team which never loses two successive ...
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0answers
15 views

Prove that the row sums of $T$ satisfy the following formula.

Consider the lower triangular matrix defined by the recurrence: $$T(n,1) = 1$$ $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = \sum _{i=1}^{n-1} T(n-i,k-1)+y \sum _{i=1}^{n-1} T(n-i,k) \; ...
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0answers
26 views

Verifying solution of difference equation?

I have the following difference equation - $2h_{x+1} - 5h + 2h_{x-1} = 0$ for $x = 1, 2, ...., 19$ The boundary conditions are $h_0 = 1$ and $h_{20} = 0$ How would I go about verifying that $h_x = ...
1
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1answer
27 views

Getting recursive formula to since solution

Is there any way to get the recursive formula of the form $r_n=\alpha r_{n-1}+\beta$ to single formula as a function of $n$. I've seen results that find single formula as function of $n$ for geometric ...
1
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1answer
49 views

Applying the master theorem

State the asymptotic runtime found by the master theorem. If the master theorem does not apply state why: 1) $T(n) = $T($n/3)$ 2) $T(n)= $ $5T$($2n/5$) + $n$ 3) $T(n) = 4T(n/2) +15n^3 + 4n^2 +n+4$ ...
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0answers
16 views

Can Banach fixed-point theorem be generalized to the case where each term depends on multiple previous terms in recurrence sequence

Banach fixed-point theorem http://en.wikipedia.org/wiki/Banach_fixed-point_theorem I'm wondering if the theorem still applies if the recurrence relation is, for example, ...
0
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1answer
44 views

proving a sequence is increasing defined by a recurrence relation.

Given the recurrence relation $b_{1}=0$ and $$3b_{n+1} = \frac{b_{n}}{12} + \sqrt{\frac{17+b_{n}^{2}}{12}}$$ Show that this recurrence relation is increasing. Note $36b_{n+1} = b_{n} + ...
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0answers
51 views

How to calculate alternating Euler sum [closed]

In How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? get $$\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2=\frac{1}{12}(\pi^2\log2-4(\log 2)^3-9\zeta(3)),$$ Similar, how to evaluate the series, ...
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2answers
39 views

recurrence problem for number of words

Let $w_n$ be the number of words (strings) of length $n$ that can be made using the digits {0,1,2,3} with an odd number of twos. Find a recurrence relation for $w_n$ and solve the recurrence. The ...
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1answer
31 views

Solving I. y[n+2]-(1/3)y[n+1]=sin(n) and II. y[n+2]+3y[n+1]-4y[n]=n-1 difference equations

I have two difference equations, which I just can't solve. I hardly even get the method, so if you could help me with the steps, I would be grateful. $y_{n+2}-\frac{1}{3}y_{n+1}=\sin(n)$ ...
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0answers
30 views

What is the growth of $T(n)=aT\left(\left\lfloor\sqrt[k]{n}\right\rfloor\right)+b\log(n)$

What is the growth of $T(n)=aT\left(\left\lfloor\sqrt[k]{n}\right\rfloor\right)+b\log(n)$ where $a$ and $b$ are positive reals and $k \ge 2$ is an integer? This is a generalization of my answer to ...
2
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2answers
76 views

Find recurrence relation of $T(n)=2T\left(\left\lfloor\sqrt{n}\right\rfloor\right)+\log(n)$

Sorry about the formatting of the title I'm not sure of the codes to make it look better. I need to find the recurrence relation of the following: $$T(0) = 1$$ $$T(n) = ...
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0answers
42 views

Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function

The reciprocal gamma function has the following Taylor series. $$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$ where the $a_k$ coefficient are given by the followint recursion. $a_1=1$, ...
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4answers
110 views

Prove sequence is Cauchy.

Prove the sequence $\{a_i\}$ defined by $a_1=1 \text{ and } a_{i+1} = 1 + \frac{1}{a_i}$ is Cauchy. And prove it converges to $\sqrt{2}$. I want to show $\lim\limits_{n\to\infty}(a_{i+1}-a_i)=0$ for ...
0
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2answers
44 views

A nice recurrence sequence

I want to find general solution for recurrence: $a_n=7a_{n-1}-10a_{n-2}+4n$ with suppose $c_1, c_2, c_3$ is constant. I need some tutorial or solution on this challenging recurrence.
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0answers
33 views

Complicated 2-variable recurrence

I am analyzing some kind of construction for which size satisfies the following recursion: $P(n,k)=\frac{k}{2}\log \frac{k}{2} + 1 + \frac{n}{2} + P(\frac{n}{2},k) + P(\frac{n}{2},\frac{k}{2})$ ...
1
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1answer
39 views

Solving recurrence equations with repeated substitution?

Say we have a recurrence equation as $$ T(n) = \begin{cases} T\left(\frac n2\right) +n & \text{if }n\ge2 \\ 1 & \text{if }n=1 \end{cases} $$ Would the first substitution be like this? ...
0
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1answer
53 views

Limit of a sequence defined by recursive relation : $ a_n = \sqrt{a_{n-1}a_{n-2}}$

We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$ $a_1$ and $a_2$ are positive constants. We have to show the following: The sequences $\{ b_n \} = \{ ...
3
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1answer
46 views

Solving recurrence -varying coefficient

How can one find a closed form for the following recurrence? $$r_n=a\cdot r_{n-1}+b\cdot (n-1)\cdot r_{n-2}\tag 1$$ (where $a,b,A_0,A_1$ are constants and $r_0=A_0,r_1=A_1$) If the $(n-1)$ was not ...
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3answers
42 views

This recurrence relation will evaluate to?

T(n) = 2T(n-1)+n, n>=2 T(1) = 1 What will this recurrence relation equation evaluate to ? I used substitution method and found out that this relation takes the form 2^k T(n-k) + 2^(k-1) * ...
2
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1answer
39 views

Solving a recurrence with a term $T(\frac n 2 + 2)$

I'm stuck trying to solve the following recurrence: $$\begin{align*} T(n) &= 4T(\frac n 2 + 2) + n : n > 8\\ T(n) &= 1 : n \leq 8 \end{align*}$$ In particular, I'm not sure how to deal ...
0
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0answers
17 views

Developing closed-form from recursive definition

Here is a recursively-defined function where c, d ∈ N. T(n) =    c, if n = 0 d, if n = 1 2T(n − 1) − T(n − 2) + 1, if n > 1 Carry out the five steps for repeated substitution to prove a ...
0
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1answer
56 views

Help with using Master Theorem on Floor/Ceiling Functions [closed]

I have to use the master theorem to find the asymptotic growth of this function in Big-theta notation. T(x) = T(⌈x/4⌉) + T(⌊x/4⌋) + √x How should I approach this ...
1
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1answer
69 views

Help with using Master Theorem on $T(n)=9T(n/3) + \Theta(n^2/\operatorname{lg}(n))$

I want to use the Master theorem to solve the following recurrence. $$T(n)=9T(n/3) + \Theta(n^2/\operatorname{lg}(n))$$ We can easily see that $a=9$ and $b=3$ and $f(n) = n^2/\operatorname{lg}(n)$. ...
3
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4answers
94 views

Recurrence relation $x_0=1, x_n=p x_{n+1} + q x_{n-1}$

I have the following recurrence relation: $$x_0=1 \\ x_n=p x_{n+1} + q x_{n-1} \text{ for }n=1,2,3,...$$ where $0<p=1-q<1$ and $0 \leq x_n \leq 1$. Edit: Sorry for the lack of context. But I ...
0
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1answer
19 views

Substitution method for solving recurrences piece wise function

I don't know how to use the substitution method for the following function: piece wise function: $T(n) = c$, if $n=0$ $T(n) = d$, if $n=1$ $T(n)=2T(n-1)-T(n-2)+1$, if $n > 1$
1
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0answers
30 views

Recurrences with modulo

Is there any specific way to solve recurrence of form (example): $$f(x) = (a \cdot f(x-1) + c) \bmod{r} $$ (when recurrence involves modulo). For recurrence without modular arithmetic I could use ...
1
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2answers
66 views

Determine whether $A215928(n)=G_n$.

Let $G_0=1$ and $G_{n+1}=F_0G_n+F_1G_{n-1}+\cdots+F_nG_0$, where $F_n$ is the $n$th term of the Fibonacci sequence, i.e., $F_0=F_1=1$ and $F_{n+1}=F_n+F_{n-1}$. Let $P_0=P_1=1,\ P_2=2,$ and ...
1
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4answers
66 views

Is this recursively defined sequence decreasing? $x_{n+1}={1\over{4-x_n}}$, $x_1=3$.

This is part of a larger problem: Prove that the sequence defined by $x_1=3$, $x_{n+1}={1\over{4-x_n}}$ converges. I want to show that it is bounded below (by $0$ or something) and that it is ...
1
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3answers
32 views

Showing that a sequence $a_n$ is a solution of the recurrence relation

I'm having some trouble with showing that a sequence $a_n$ is a solution to the recurrence relation $a_n = -3a_{n-1} + 4a_{n-2}$. (See image below). The sequence $a_n$ that is given $= (-4)^n$ . I'm ...
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0answers
19 views

How to compute this Z-Transform?

The exercise is like this: $$y(k+1) - 3y(k) = 4^k$$ How do I compute $Z$ transform of $4^k$? I understand that I have to use the Z-Transform formula and the result after applying it is : $$sum ...
2
votes
3answers
57 views

Proof by induction that $f(n) = 1-2^{2^n}$, where $f(0) = 3$ and $f(n) = 2 f(n-1) - (f(n-1))^2$

I am doing a textbook question which state that a function $f:\mathbb{N}\to\mathbb{Z}$ is a recursively defined as shown bellow $f(0) =3$, $f(n) = 2\cdot f(n-1) -(f(n-1))^2 $ if $n\ge1$. Prove that ...
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2answers
33 views

How to analyze convergence of non-linear difference equation (recurrrence relations)

I've a couple of functions, such as: $Y(t+1)=2-\ln(Y(t))$ $Y(t+1)=(Y(t))^{-2}$ $Y(t+2)=e^{-Y(t)}$ and I need to analyze stability and convergence. No problem with stability, but I can't figure out ...
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3answers
275 views

Show that the greatest common divisor of any two terms in recursive sequence $T_{n+1} = T_n^2 - T_n + 1$ is 1.

Question: Consider the sequence of integers $T_n$ , where $n ∈ N$ defined by the recurrence relation $T_0 = 2$ , $T_{n+1} = T_n^2 - T_n + 1$ (for $n ∈ N$). Show that for any $m$ , $n ∈ N$ ...
1
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1answer
47 views

Dynamic Programming - how to minimize sum of distances

Let's assume that we're given the num[N], an array of N positive integers in an ascending order. For instance, let's assume that N=10, and num[N] is the following: 1 2 3 6 7 9 11 22 44 50 Let ...
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0answers
32 views

How to work with a recursive function with 2 recursive instances?

In class, we figured out how to find the closed form of a recursive definition through the "basic 5 steps method". Example function T(n): If n = 1, T(1) = 1 If n > 1, T(n) = T(n-1)+1 Step 1: ...
0
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1answer
50 views

Finding recurrence relation on a problem

I need a little bit help finding a recurrence relation. So it goes like this: "A one-sided pavement is being made with tiles that come in 5 different colors. There are 3 light colors (light-yellow, ...
19
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1answer
404 views

Infinite staircase to a circle

Suppose you start at $(0,0)$ on the unit disc and repeat the following procedure again and again: Face east and walk half-way to the circumference. Face north and walk half-way to the circumference. ...
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2answers
36 views

How do I solve this recurrence relation?

Given a recursive relation $$a_n = \begin{cases} (1 - 2b_n)a_{n-1} + b_n, & n > 1 \\ \frac{1}{2}, & n =1 \end{cases} $$, how can I expression $a_n$ in term of $b_i, i \in \{1, 2, \dots ...
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1answer
23 views

Checking recurrence relation

Is there a way to check my recurrence relation, so I can confirm I did it correctly? $a_k = -4a_{k-1} -4a_{k-2}$ with $a_0 = 0$ $a_1= -1$ My answer: $a_n = 0(-2)^n - ½n(-2)^n$
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1answer
36 views

How do I use complete induction here?

Suppose currency consists of 3 and 4 cent coins. Suppose you want to buy an item that is worth 9 cents. Show that if you have an unlimited number of 3 and 4 cent coins you can buy anything greater ...
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0answers
14 views

Maxima of a recurrence

The following recurrence has a maxima around $k = \lceil \log_d{n}\rceil$, where $n > 0$, $d > 3$: $$b(n,k) = {b(n-1,k-1)\over {d^{k-1}}} + (1-{1\over {d^{k}}})b(n-1,k)$$, where $0 \le k \le ...
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0answers
37 views

combinatorial maths recurrence relations for distribution of N objects into 3 different boxes

Given N balls which are red, green or blue in color.And three boxes R,B,G.In how many ways these N balls can be distributed so that box R contains atleast r no of red balls, B contains atleast b no of ...
6
votes
1answer
148 views

permutation and f(n) challenge

Suppose $f(n)$ be the number of permutation from set ${1,2,..,n}$ such that for each $ 1 \leq i \leq n$ we have: $ | \pi(i)-i| \leq 1 $. meaning of $ \pi(i)$ is an elements whose in place $i$ of ...
0
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0answers
42 views

Solving a linear recurrence with unknown changing coefficient.

I'm stuck on how to solve this recurrence (if it can be solved?) Any help or tips would be greatly appreciated. \begin{equation} x_n=a_nx_{n-1}-x_{n-2} \end{equation} with $x_1=-1$ and $x_2=-a_2$ ...
2
votes
2answers
38 views

2 element subsets of n elements?

the question is as follows: Give a recursive definition for the number of $2$-element subsets of $n$ elements. We started working this out in class and here is where we got too: -if $n = 0$, then ...
2
votes
1answer
64 views

Do we have to claim it? If so, at which point?

I have to solve the recurrence relation $$T(n)=\left\{\begin{matrix} 3T\left (\frac{n}{4} \right)+n & , n>1\\ 1 &, n=1 \end{matrix}\right.$$ and prove by induction that the solution I ...
1
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1answer
19 views

Identifying R1 and R2 when solving Recursion relations

We are learning to solve recursion relations. When I get this step, does it matter if I define $r_1$ as 5 or 2 in this example?
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0answers
11 views

Discrete Analoge Methods for solving difference equations

For solving non-linear first order differential equations we can use separation of variables (sometimes) or an integrating factor to convert a DE to an exact DE. Are there any analog methods for ...