Recursion is the process of repeating items in a self-similar way. A recursive definition (or inductive definition) in mathematical logic and computer science is used to define an object in terms of itself. A recursive definition of a function defines values of the functions for some inputs in ...

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What Is The Next Best Ingredient?

I am building a program that allows users to optimize their grocery shopping so they can make the most recipes using the fewest ingredients. One of the features of this program is a function I’m ...
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72 views

Finding the convergent value of a recursion similar to Arithmetic-Geometric Mean recursion

The sequence is defined as follows : Start : $(x_0,y_0)$ with $ 0 < x_0 < y_0 $ Step : $x_{n+1} = \frac {x_n+y_n} {2}$ , $y_{n+1}= \sqrt{x_{n+1}y_n} $ Find $\lim_{n\to \infty}(x_n,y_n)$ . ...
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Relation between 2 recurrence equations.

I am stuck with the following recurrence relations (actually asked in a programming question).Given the following relationships, is it possible to find the index of the occurrence of any given number ...
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2answers
38 views

8th positive odd integer that is an ODD Catalan number? [on hold]

The $n^{\text{th}}$ Catalan number is given by the formula $C_n = \frac 1{n+1}\binom{2n}n$. It also satisfies the recurence \begin{align*}C_n &=\sum_{k=0}^{n-1}C_kC_{n-1-k}\\ &= ...
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Recursive Integration over Piecewise Polynomials: Closed form?

Is there a closed form to the following recursive integration? $$ f_0(x) = \begin{cases} 1/2 & |x|<1 \\ 0 & |x|\geq1 \end{cases} \\ f_n(x) = ...
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1answer
58 views

How to approach an equation of the form $f(x)=y$ where $f$ is recursively defined? [duplicate]

This is a homework problem so I am not looking for someone to solve it for me. I would like to know how I should approach this problem, or in what direction I should research to figure it out myself. ...
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4answers
34 views

General Solution for a non-Homogeneous Recurrence

What is the general solution to the recurrence: $$x(n + 2) = x(n + 1) + x(n) + n - 1$$ where $n\geq 1$ with $x(1) = 0, x(2) = 1$? It's a question on a practice exam I'm reviewing and I'm not ...
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1answer
19 views

Recursive functions.

If you have a recursive function $$g(x) = f(f(x))$$ and you know that $$f(0) = 0, f'(0) = 1, f''(0) = 2$$ Will then $$g(0) = 0, g'(0) = 1, g''(0) = 2$$ ?
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1answer
35 views

Express recurrence in closed form

I am having trouble understanding the process of expressing the following recurrence in its' closed form. First of all, I do not really understand what "closed form" means. If someone could elaborate ...
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3answers
25 views

Proof for result of sum of 3 elements of recursive sequence

I have a recursive sequence: $$a_1=1\\a_2=1\\a_3=-1\\a_k=a_{k-1}\cdot a_{k-3} (for\,k>3)$$ So this sequence has cycle of 7: $1,1,-1,-1,-1,1,-1$ And I have to calc $a_{2013} + a_{2014} - ...
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1answer
18 views

Proof by induction for a recursive function f

Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive case: $\operatorname{f}(x) = ...
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1answer
38 views

How can I find the order of growth of this recurrence: $T(n) =\sqrt{n}T\bigl(\sqrt{n}\bigr) + n$

I am trying to find the order of growth ($O(n)$, $O(n\log n)$) of the recurrence $T(n) = \sqrt{n} \cdot T\bigl(\sqrt{n}\bigr) + n$. I started to unroll the recurrence and found that I can rewrite it ...
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1answer
23 views

Matlab Recursion Loop

I need to create a loop to answer part c of the following linear algebra question. I'm new to Matlab so this is extremely confusing. I'm not sure how to make a loop that "counts" from 0-25 in order to ...
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1answer
26 views

Proof by induction for a recursive function

Really having a tough time doing this question: Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive ...
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3answers
58 views

Constructing a recursive sequence that converges to sqrt 17

One of the problems that we have for abstract math is the following: Using the recursive sequence definition, construct a sequence that converges to $\sqrt{17}$. It is my understanding that the ...
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0answers
11 views

largest number of regions formed by a certain amount of planes recursion question

The question: What is the largest number of regions formed by $6$ planes in space? I answered a similar question to this and it said "What is the largest number of regions formed by $6$ LINES in ...
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1answer
43 views

$T(n) = \sqrt n\,T(\sqrt n) + n\log n$

i tried to solve this recursion equation with master theory, and its not working in this way. How many arrays exist in each steo in recursion tree? And how can i solve this problem with another way? ...
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1answer
107 views

All solutions of the recurrence relation

Find all solutions of the recurrence relation $$ a_n = 2a_{n-1}+15a_{n-219}-64a_{n-3}+k $$
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2answers
264 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: \[ g(1)=\alpha \] \[ g(2n+j)=3g(n)+\gamma n+\beta_j \] \[ \mbox{for}\ j=0,1\ \mbox{and}\ n\geq1 \] I ...
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1answer
311 views

The problem of the most visited point.

Represents the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ as points in the figures below for exemple. How to calculate the number of circuits that visit ...
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0answers
6 views

QBF - space complexity in detail

As I'm new to the "complexity theory" stuff I've some trouble with proofs which are "obvious" regarding all books I've found so far. In this case I want some evidence why a certain algorithm has space ...
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0answers
6 views

Dinamic programming for sum of two largest values from a Uniform parent

I must find a recursion formula to find the sum of two largest values from a Uniform parent [0,1]. I need dinamic programming but I don't know how to organize it.
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1answer
35 views

How to solve a recursively defined system

It has been a while since I have tackled a problem like this and could use a refresher. I have a recursive system of equations that looks a little like: Where the initial x & y values are ...
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How to prove $\exists f (x,y)$, $g(x,y)$ such that $W_{ f(x,y)} =W_x \cup W_y$ and $W_{g(x,y)}=W_x \cap W_y$

According to S-m-n Theorem there are primitive recursive functions $r(x,y)$, $s(x,y)$ such that $$W_{u(x,y)}=W_x\cup W_y,\\W_{i(x,y)}=W_x\cap W_y.$$ How to prove there exist $f (x,y)$, $g(x,y)$ ...
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1answer
14 views

Log property in proof of Master theorem

The family of recurrence considered is of the form $$ T(n) = aT(n/b) + n^c $$ $a,b,c$ are integers. One case of master theorem states: if $c< log_b a$, then $T(n) = \Theta(n^{log_b a}) $. I have ...
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2answers
23 views

Is there solution for equation which is recursive?

I have this following vector equation:$$\vec x = \vec x_0 + (1/2) * f(\vec x, t)t^2$$ where $\vec x$ has initial value when first fed into function $f$. All vectors are 3 dimensional and function $f$ ...
2
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2answers
356 views

Primitive recursive definition of the “divisibility” relation

Let $$d(x,y)= \begin{cases} 1, &\text{if }x\text{ is divisible by }y \\ 0, &\text{otherwise.} \end{cases}$$ How can I define $d(x,y)$ in terms of just the basic primitive recursive functions ...
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1answer
37 views

Solution of recurrence

I need some explanations at the proof of the following theorem. Theorem: Let $a$, $b$ and $c$ be nonnegative constants. The solution to the recurrence $$T(n)=\left\{\begin{matrix} b & ,\text{ ...
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1answer
33 views

A general or simple method to solve this iterative/recursive problem?

I have the following iteration $$ y_n\mapsto n\sum_{m=1}^{n+1}y_m $$ starting with $y_1$ being a positive real number. Is there a standard method to find the coeffcients of all $y_n$ after ...
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1answer
16 views

Find a closed form equation of the following sequence: ${0,0,-2,0,4,0,-6,…}$

Find a closed form equation of the following sequence: ${{0,0,-2,0,4,0,-6,...}}$ I know $1+-1^n$ = 0 if n is odd and 1 if n is even. However finding alternating signs when plugging in only even ...
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2answers
47 views

How does mathematics fit into fractal generation for computer graphics?

I have to do a research paper on any mathematical concept. The mathematical concept must be complex, so I thought fractals would be a good choice (I was told it was a complex idea). I have been ...
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1answer
31 views

Conditional iterations constant.

Let $f(0)=2.$ Define for positive integers $n$ : $f(n+1) = \frac{3}{2} f(n)$ if $f(n)$ is even. $f(n+1) = \frac{3}{2}(f(n)+1)$ if $f(n)$ is odd. We now have $\lim_{n->\infty} \dfrac{4* (3/2)^{n} ...
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1answer
65 views

Proving $V=V_{\sf On}$ in $\sf Z+Reg$

Define the function $$V_0=\emptyset,\qquad V_{\alpha+1}={\cal P}(V_\alpha),\qquad V_{\delta}=\bigcup_{\beta<\delta}V_\beta.$$ Since we are working in $\sf Z$ (i.e. $\sf ZF$ without the axiom of ...
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1answer
16 views

How can i find the complexity of this recurrence relation?

Basically i'm having this recurrence relation which i don't know how to get the complexity of it by using the iterative method $T(n) = \begin{cases} 0, & \text{if $n=0$} \\ 1, & \text{if ...
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2answers
115 views

Recursion on a class instead of a set

According to Wikipedia, the recursion theorem states the following: Let $X$ be a set, and $f:X\to X$ a function. For any $x\in X$, there exists a unique function $g:\omega\to X$ such that $g(0)=x$ ...
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48 views

Discrete maths - Recursion, formula and induction question

A sequence $S_{0},S_{1},\ldots$ is defined recursively as follows: $$ S_{0} = 3\,,\quad S_{k} = S_{k - 1} + 2k\quad\mbox{for}\quad k \geq 1 $$ Calculate a few terms of the sequence and conjecture a ...
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1answer
25 views

Solution for $T(n) = 2T(\sqrt{n}) + log_2(n)$ [closed]

Solve for: $T(n) = 2T(\sqrt{n}) + log_2(n)$ with no base conditions.
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1answer
251 views

Closed form for general recursive function

Does a closed form exists for general recursive functions? my guess is not, but what types can be solved or what are the constraints on a recursive function so it has a closed form, what are some ...
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0answers
25 views

Financial mathematics: asion option

I've got a forward starting asian call option: $V_N = max\left( 0,\left(\sum_{j=m+1}^{N}{S_j} - K \right) \right) $ So the payoff is determined by the average stockprice over de latest N-m days and ...
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1answer
33 views

Solution to recursion equation

Solve the recursive equation $$ T(n) = T(n-1) * T(n-2) $$ with $T(1) = a$, $T(2) = b$ How do I solve this algebraically? I unrolled the recursion and got a solution of $a^{f_n}b^{f_{n+1}}$, ...
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1answer
47 views

How to find explicit formula using backwards iteration

For this problem I tried to solve it by iterating downward so: an = an-1 - n an-1 = 2*(2an-2 - (n-1)) - n = 4an-2 - 3n + 2 an-2 = 2*(4an-3 - 3n + 2) - n = 8an-3 - 7n + 4 an-3 = 2*(8an-4 - 7n + ...
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1answer
25 views

How to prove that if $T(n)=\max_{1\le q \le n-1}(T(q)+T(n-q))+\Theta(n)$ then $T(n)=O(n^2)$

I have this recursion relation: $T(n)=\max_{1\le q \le n-1}(T(q)+T(n-q))+\Theta(n)$. How do I show that: $T(n)=O(n^2)$ Thank you!! (P.S. I hope it's OK to ask it, but I'm looking for the simple ...
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24 views

Ways of defining a recursive function that counts right-parenthesis in a string

I'm trying to find a more elegant way of defining a recursive function on $\{(,)\}$ that counts right-parenthesis in a string. Let r be a function on $\{(,)\}$ defined recursively so that: ...
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1answer
21 views

How to solve $T(n)=\frac2n\sum_{i=1}^{n-1}T(i)+\Theta(n)$

I need to solve this Recursion Relation:$$T(n)=\frac2n\sum_{i=1}^{n-1}T(i)+\Theta(n)$$ Do you have any idea how? (I try to find another way instead induction). (We should get: $T(n)=\Theta(n\log n)$) ...
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2answers
34 views

Solving recurrence relation $f(n) = f(\lfloor\sqrt n\rfloor) + 1; f(1) = 1, f(2) = 1$

As the title shows, I need help approaching a solution for recurrence relation: $f(n) = f(\lfloor\sqrt n\rfloor) + 1$ if $n\ge3$ with initial values $f(1) = 1$, $f(2) = 1$ I am particularly ...
0
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2answers
34 views

Using Master Theorem to prove a recurrence with f(n) = Θ(n/logn)

I'm trying to use the master theorem to solve the recurrence: $$T(n) = 4T\left(\frac{n}{5}\right) + \Theta\left(\frac{n}{\log n}\right)$$ I'm having trouble understanding how the ...
2
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2answers
62 views

how many even number faced dice will be their if 99 dices roll for eternity with some condition

Consider a six ­sided dice with number from 1 to 6. Imagine you have a jar with 99 of such dices. You throw all dices on the floor so they all land at different numbers. You look at one dice at a ...
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14 views

Complexities of Recurrences in the form $t(n) = t(\alpha n) + t(\beta n) + cn$

When considering recurrence relations, they are generally of a form similar to $$t(n) = t(\alpha n) + t(\beta n) + cn$$ and there are three cases to be considered for our values of $\alpha$ and ...
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0answers
11 views

Recursion and Time Complexity Concept

I have been solving this question which appeared in one of the entrance exam. The question is as follows: ...
1
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2answers
32 views

Recursive Equation.

We have a recursive function: $$a_n = Aa_{n-1} + Ba_{n-2} $$ We assume that $ x ^ n = a_n $. From this equality quadratic equation we have two solutions $$ \alpha, \beta, \qquad\alpha \neq \beta $$ In ...