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 solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?

This looks like a question asked earlier, but it isn't. $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad & \text{ if } n>1 \\ 1 & \text{ if }n=1\end{cases}$$ My professor gave this to me in ...
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1answer
338 views

Establishing formula from recurrence

Can anyone tell me how do we establish a formula from a given recurrence relation? Take the example of $f(n) = 2f(n-1) + 1$, $n \in \mathbb{Z^+}$, $f(1) = 1$ When I write out the first few values, it ...
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4answers
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solve $T(n)=T(n-1)+T(\frac{n}{2})+n$

Using the recursion tree i tried solving this: $T(n)=T(n-1)+T(\frac{n}{2})+n$; the tree has two parts (branches) one that of $T(n-1)$ and other branch is of $T(\frac{n}{2})$. But as the term T(n-1) ...
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5answers
183 views

How to get nth derivative of $e^{x^2/2}$

I want to calculate the nth derivative of $e^{x^2/2}$. It is as follow: $$ \frac{d}{dx} e^{x^2/2} = x e^{x^2/2} = P_1(x) e^{x^2/2} $$ $$ \frac{d^n}{dx^n} e^{x^2/2} = \frac{d}{dx} (P_{n-1}(x) ...
13
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1answer
301 views

Primitive recursive function which isn't $\Delta_0$

What is the simplest/cutest example (and/or example with the most student-friendly proof that it is an example) of a primitive recursive function which isn't representable by a $\Delta_0$ wff?
12
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3answers
924 views

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) = ...
10
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2answers
632 views

The set that only contains itself

Ignoring the axiom of regularity (and therefore the implication of "no set can contain itself"), would it be correct to state that the set that contains only itself is unique? My argument is that if ...
0
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1answer
354 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
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2answers
124 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 ...
3
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2answers
224 views

Proving a recurrence relation for strings of characters containing an even number of $a$'s

We consider strings of $n$ characters, each character being $a$, $b$, $c$, or $d$, that contain an even number of $a$'s. (Recall that $0$ is even.) Let $E_n$ be the number of such strings. ...
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3answers
60 views

First order difference equation. Solve $u_{n+1}=3u_n+2$.

First order difference equation. Solve $u_{n+1}=3u_n+2$ with $u_0=0$ My notes are very sparse on this topic, so I need some help solving what should be an easy question. I would really appreciate ...
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1answer
86 views

Recursive Function - $f(n)=f(an)+f(bn)+n$

I've got this recursive function $f(n)=f(an)+f(bn)+n$, and I need to find $θ$ on $f(n)$, as $a+b>1$. Using a recursive tree, I managed to bound it by $n\log(n)$ from the bottom and by $n^2$ from ...
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3answers
109 views

Proof by induction that recursive function $\text{take}$ satisfies $\text{take}(n) = 100 - 2n$

I'm sick and tired off posting threads about induction... I just can't seem to get it, I need someone to give me a detailed explanation and treat me like a 5 year old, literally. I'm wasting a lot of ...
0
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2answers
83 views

Possible distinct binary tree structures at depth d

I'm trying to figure out a recursive formula for the number of possible distinct binary trees at any depth d. I haven't been able to find any sort of sources on this. basically, at depth 0, the only ...
0
votes
1answer
72 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 ...
0
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1answer
67 views

How many combinations can I make?

let $n \gt 1$ be an integer, and consider $n$ people; $P_1, P_2,..., P_n$ let $A_n$ be the number of ways these $n$ people can be divided into groups, such that each group have either one or two ...
0
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1answer
284 views

Simultaneous recursion

I have no idea how to even start proving the following theorem: If $f_0, f_1: \mathbb{N}^r \rightarrow \mathbb{N}$ and $g_0, g_1: \mathbb{N}^{r+3} \rightarrow \mathbb{N}$ are primitive recursive, ...
0
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2answers
2k views

Asymptotic bounds of $T(n) = T(n/2) + T(n/4) + T(n/8) + n$

This problem is given in "Introduction to Algorithms", by Thomas H. Cormen. I have the answer to it, but I don't understand it. The answer is, $T(n) = \Theta(n)$. It would be really good if you ...
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1answer
462 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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3answers
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Is there a slowest divergent function?

So I've been playing around with some functions for a while, and started wondering about a slowest divergent function(as in $\lim_{x\to\infty} f(x)\to\infty$) and so I searched around for an answer. ...
11
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1answer
119 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
8
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1answer
631 views

Bell numbers, recursions, bijections

The $n$th Bell number, named after Eric Temple Bell (although he was far from the first to think about them), is the number of partitions of a set of cardinality $n$. If they are written in the top ...
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5answers
714 views

Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$)

The Problem How can you evaluate (i.e., get a value for) Tetration (i.e., iterated exponentiation) to infinite heights? For example, what would be the value of this expression? $$ ...
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3answers
672 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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186 views

Recursive and Primitive recursive functions

According to the book that I'm reading, we can define the $\mu-$recursive functions inductively, as follows: The constant, projection, and successor functions are all $\mu-$recursive. If $g_1, ...
6
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1answer
97 views

Eigenvalues appear when the dimension of the Prime Index Matrix is a prime-th prime. Why?

I had a look at the eigenvalues of the matrix, I called it Prime Index Matrix (is there a better name?), constructed like the following: $$ P_{k,p_k}=P_{p_k,k}=1, $$ where $p_k$ is the $k$th prime. ...
5
votes
2answers
418 views

A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of ...
3
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2answers
56 views

Number of $1$s in the binary representation of $n$

Trying to define the function $b(n)$ which counts the number of $1$s in the binary representation of $n$ arithmetically I came up with the following definition: $$b(n)=m :\equiv (\exists k_1\dots ...
6
votes
2answers
100 views

Prove function is Fibonacci sequence

Stuck on how to finish a question I'm working on. Have to find the number of bitstrings of length n with no odd length maximal runs of ones. For example, when n=3 there are three such bitsrings: 011, ...
5
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1answer
222 views

Do we really need the recursion theorem if we deal only with specific recursively defined functions?

How is it possible to define in a totally rigorous (i.e. from the axioms) was the functions $$h:\mathbb{N}\rightarrow \mathbb{N}, \ n\mapsto 1\cdot\ldots \cdot n$$ or $$ g:\mathbb{N}\rightarrow ...
4
votes
1answer
105 views

Does every positive rational number appear once and exactly once in the sequence $\{f^n(0)\}$ , where $f(x):=\frac1{2 \lfloor x \rfloor -x+1} $

Consider the map $f:\mathbb Q^+ \to \mathbb Q^+$ defined as $f(x):=\dfrac1{2 \lfloor x \rfloor -x+1} , \forall x \in \mathbb Q^+$ ; then is the function $g:\mathbb Z^+ \to \mathbb Q^+$ defined as ...
4
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1answer
96 views

An effective coding of $\mathbb N^*$

Problem: Assume $\Pi:\mathbb N^2\to\mathbb N\setminus\{0\}$ is a primitive recursive coding of the pairs of numbers, that is also a bijection and $(\forall (x,y)\in\mathbb N^2)(\Pi(x,y)>max(x,y))$. ...
4
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1answer
107 views

Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that ...
3
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1answer
86 views

Solve a recursion using generating functions?

Given the recursive equation : $$F_n+F_{n-1}+⋯+F_0=3^n , n\geq0$$ A fast solution that I can think of is placing $n-1$ instead of $n$ , and then we'll get : $$F_{n-1}+F_{n-2}+⋯+F_0=3^{n-1} $$ ...
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0answers
76 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
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1answer
35 views

Finding permutations recursively.See constraints below

Problem: Let $P(n)$ be the number of permutations of $m$ letters taken $n$ at a time with repetitions but no $3$ consecutive letters being the same. Find a recurrence relation connecting $P(n)$, ...
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1answer
285 views

All infinite recursive sets can be enumerated by an injective function

Definition: We call a set recursive if its characteristic function is recursive. Claim: If the recursive set $A\subset\mathbb N$ is infinite then $A=Range(f)$ for some primitive recursive injective ...
0
votes
1answer
90 views

A setting in which Rice's theorem is not true

In my class we call a set of computable functions $A$ recursive if its indexing set $I_A=\{e\in\mathbb N:\phi_e\in A \}$ is recursive, where $\phi$ is some known Gödel numbering of the computable ...
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3answers
281 views

(CHECK) $n$-bit Strings Containing a Pattern

$$\text{$\bf{PLEASE~~~CHECK~~~AUTHOR'S~~~ANSWER}$}$$ If $S_n$ denotes the number of $n$-bit strings that do not contain the pattern $00$, then what is the underlying recurrence relation and ...
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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 ...
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Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ...
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Where do the first two numbers of Fibonacci Sequence come from? [duplicate]

I'm trying to code a simple algorithm that prints out the $n^{th}$ Fibonacci number. However, my program requires the initial seed values $F_0 = 0$ and $F_1 = 1$, when I'm hopeful I can figure ...
4
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2answers
146 views

$ T(n)= T(\log n)+ \mathcal O(1) $ Recurrence Relation

what is the solution of following recurrence relation? $$\begin{align} T(1) &= 1\\ T(n) &= T(\log n) + \mathcal O(1) \end{align}$$ a) $O(log n)$ b) $ O (log^* n) $ c) $ O (log^2 n) $ d) $ ...
4
votes
4answers
379 views

Proving convergence of a sequence

Let the following recursively defined sequence: $a_{n+1}=\frac{1}{2} a_n +2,$ $a_1=\dfrac{1}{2}$. Prove that $a_n$ converges to 4 by subtracting 4 from both sides. When I do that, I get: ...
4
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2answers
12k views

Proof by Induction for a recursive sequence and a formula

So I have a homework assignment that has brought me great strain over the past 2 days. No video or online example have been able to help me with this issue either and I don't know where to turn. I’m ...
4
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1answer
108 views

Recursively Defined Entities

So I am having some trouble understanding how one is to come up with the recursive definition to the following problem... We are given a rectangle of width $2$ and length $n$. Suppose we have ...
3
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1answer
38 views

Prove $A(x,y)= 2[x](y+3)-3$. Where A is the Ackermann-Peter function and [x] is x-th hyperoperator.

I've successfully proven $A(x,y)$ for some fixed x and any y with induction but I'm having a hard time proving this for any x and y. I think the next useful step would be proving $A(x,0)= 2[x]3-3 $ ...
3
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3answers
161 views

Counting the number of different ways in which groups of one or two can be formed…

I'm having trouble proving that the number of ways n>3 people can be divided into groups of either one or two is equal to: $A_n = A_{n-1} + (n-1)⋅A_{n-2} $ I'm trying to prove this by counting but ...
3
votes
2answers
3k views

Convergence and limit of a recursive sequence

Let $p>0$ and suppose that the sequence $\{x_n\}$ is defined recursive as $$ x_1 = \sqrt{p}, \quad x_{n+1} = \sqrt{p + x_n}, $$ for all $n \in \mathbb{N}$. How can I show that $x_n$ converges, ...
3
votes
3answers
147 views

Can I use the master theorem for this?

this is a HW question so please don't just give me the answer right away. Basically, I'm working on solving the running time of this recurrence method: $$T(n) = 4T(n/3) + n \log \log n$$ I want to ...