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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n-th number with given prime divisors

I would like to compute th $n$-th positive integer whose prime divisors are among numbers $2$, $3$ and $5$. $n$ is at most $12500$. My first approach was sieving but i found out that there are less ...
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Finding general formula for a sequence that is not arithmetic and neither geometric progression?

I have this $$a_{n+1} = a_n + 4n - 1\qquad a_1 = 2$$ And I need to find general formula for $a_n$. This is one of the last exercises for the question related to it so I'll give a summary of what I ...
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Converting Non-linear Recursive Series into Explicit Form

I know it's possible to convert any (I think it's any, at least) first-order recursion into an explicit form. For example (assuming I did this right): ...
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Probability Method Of Recursive number patterns in everyday things

I'd like to know how to go about calculating the probability of a recursive number pattern happening in everyday things for an essay which I'm writing. The example of a recursive number pattern would ...
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58 views

Fractal fundamentals

I am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and ...
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Plotting y=x^(1/y)

So I was toying around with the idea of recursive functions, where there are two variables, and one of them is on both sides of the equation. I stumbled upon/came up with this function: y = x^(1/y), ...
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A semi-recursive infinite set is the range of some injective recursive total function

The wikipedia article for semi-recursive sets (formally titled "recursively enumerable sets") claims: A set S of natural numbers is called recursively enumerable if there is a partial recursive ...
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18 views

Showing the number of y < x such that xRy is primitive recursive.

Suppose that $Rxy$ is a (primitive) recursive relation. Let the function $ \phi $ be defined as follows: $\phi(x)$ = the number of $y < x$ such that $Rxy$. Show that $ \phi $ is ...
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34 views

how do I prove this by induction? (recursion)

The terms are given recursively: $P_0=3$ $P_1=7$ and $P_n = 3P_{n-1}-2P_{n-2}$ for $n\ge2$ What should I assume and what step proves that $P_n=2^{n+2}-1$ is a closed form of the sequence. Suppose ...
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44 views

Find and solve a recurrence relation for the number of matches played in a knockout tournament with $n$ teams, where $n$ is a power of $2$

Find and solve a recurrence relation for the number of matches played in a knockout tournament with $n$ teams, where $n$ is a power of $2$ I have already found a recurrence relation based on the two ...
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30 views

I have a proposal on recursive functions, but I need a second voice to be sure it makes sense.

My main field is computer science. Although I am mostly familiar with computer science theory, I lack in math theory. So, to an average computer scientist, the term ‘recursive function’ will invoke ...
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46 views

Finding 1996'th term for given recurrence relation

I've been given the sequence $a_n$ defined by the recursion relation: $$a_{n+1}=\frac{a_n}{1+na_n}\qquad a_1=1$$ and have been tasked to find $a_{1996}$. How would I go about that? I have a basic ...
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27 views

Showing a relation is primitive recursive, recursive, or semirecursive.

I am not sure what strategy to use to I should use to show this is primitive recursive. I believe I am to show all three cases: primitive recursive, recursive, and semi-recursive. The diagonal of ...
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1answer
19 views

Induction on a Recursive Sequence?

So I don't really know where to go from here, or how to "guess a formula for an" a0, a1,a2... is a sequence that a0 = a1 = 1 and, for n >= 1, an + 1 = n (an +an-1) So I started off by doing the base ...
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40 views

Recursive Definitions

I have two recursive definitions that I need to determine if they are correct. The first one is: The set S of all positive integer multipliers of 5 can be described by the following recursive ...
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34 views

Is math recursive or iterative?

Is the process of solving a mathematical problem (algebraic equations, limits, derivatives, integrals, EDOs, trigonometric identities proof) recursive or iterative? For example, for solving $x=1+2+3$ ...
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17 views

Iterated Function System for Cosine

So, if we let $f_1(x) = 2x^2-1$ and say $f_{n+1}(x) = f(f_{n}(x))$ then some graphing and experimenting seems to indicate that $\lim_{n\to\infty} f_n(2^{-n}x) = \cos(x)$. They converge pretty quickly ...
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How to define a set of trees recursively?

In particular, consider the set of integer-labelled binary trees (T). How could this set be defined in a recursive way from $\mathbb Z$ and T itself? Examples: $(-2, 1, (3, 1, 0)) \in T$ $(-1, (7, 2, ...
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22 views

Troubleshooting Textbook: Using Generating Functions for Non-Homogenous Recurrence

I am learning about Generating Functions to solve non-homogenous linear recurrences, and I can't seem to get the right solution, no matter what I do. Also, the example I have to work off of in the ...
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25 views

Need help finding general formula for an iterative equation.

If we have a iterative equation defined as for $i=1:n$ $$a=a(4-a),$$ I need help finding the general formula for this in terms of n. I know some obvious ones like, $$n=1\quad\quad a=4a-a^2$$ $$n=2 ...
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44 views

Prove a Recursive Formula by Induction?

So I have a bonus question on a homework assignment I am working on that literally just asks "How would you prove a recursive formula by induction?" There are no numbers, or sequences given. I ...
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26 views

Do I have these recursive and closed forms correct?

For the sequence: $0,1,5,12,22,35,51,70,92,117,145,176$ I have the closed form (dashes indicate subtext): $$a_n=\frac{n(3n-1)}{2}$$ For recursive: $$a_{n+1}=\frac{a_n+3n^2+5n+2}{2}$$ If they are ...
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given an integer i and a range r of real numbers, i divides ranges exactly in half sequentially, where does i split the next r? (without recursion)

Say I have a range r from a - b, $a\in \mathbb{R}$ and $b\in \mathbb{R} $. Then, I have an integer i. Now, if I were to break the range in half, decrease i, then continue breaking the remaining ...
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52 views

Material with half-life that is being added to periodically

I am having trouble calculating this problem: $X$ amount of material with $Y$ half-life is administered on a patient. When an interval of time $=A$ elapses, ( as a given fraction of half-life) an ...
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158 views

Find Nth formula of recursive formula $a_n=a_{n-1}+n(n-1)a_{n-2}$

$$a_n=a_{n-1}+n(n-1)a_{n-2}$$ $$a_0=1, a_1=-\frac{1}{2}$$ Is it possible to find explicit formula for $a_n$ just by using $a_0$ and $a_1$? I know how to solve this problem if $a_n=Aa_{n-1}+Ba_{n-2}$ ...
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How can I solve this linear recursion?

Let be $a_{0}=1,a_{1}=1,a_{n}=4a_{n-1}-2a_{n-2}$ if($n\ge 2)$ Should I first find the generating function of the recursion and after that? I solved it with Wolfram Alpha and after it the result is ...
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16 views

Recursion If $a_0=1$, $a_1=3$, $a_2=9$ and $a_{n+3}=a_{n+2}+4a_{n+1}+5a_n$, show $a_n\le 3^n$

If $a_0=1$, $a_1=3$, $a_2=9$ and $a_{n+3}=a_{n+2}+4a_{n+1}+5a_n$, show $a_n\le 3^n$. I don't know how to type it in right format. $n+3$ and such things in the parentheses are small and in the lower ...
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Define algorithm using divide and conquer paradigm [closed]

Q:Describe a Θ(n lg n)-time algorithm that, given a set S of n integers, determines which two elements in S have the smallest difference. (From what i understand, we first apply merge sort to our ...
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recurrence algorithm unfolding with answer, need explanation

SO the question is to prove $O(n\log n)$ hence figure out $f(n)$ correct? why does $n = 2^{2^k}$ what is so special about $2^{2^k}$ can i use another number? this answer is using substitution? I ...
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23 views

equivalent definitions of recursively enumerable sets

In some textbooks, a n-ary set R is defined as r.e iff there's it is a domain of a recursive function. In others, definition is restricted to case n=1 and a set is called r.e. if it is a range of ...
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Characterization of $\Delta^0_0$ (rudimentary) functions

A $\Delta^0_0$, or rudimentary, functions $\Bbb N^k \rightarrow \Bbb N$ is a function whose graph is definable by a bounded formula. Can this class of functions characterized by means of closure ...
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Derivation of tetration by iteration

I was screwing around a bit differentiating tetrations and was trying to write some rules for them. I came up with this recursive definition: $$ \frac{d\ ^nt}{dt} =\ ^nt \cdot log(t)\frac{d\ ...
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44 views

Recursive formula for a visual pattern

I was looking at some of the examples at visualpatterns.org and coming up with explicit and recursive formulas for various aspects of the patterns. Consider the pattern below and the number of cubes ...
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84 views

Example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive?

Browsing wikipedia, I stumbled on the following: "The image of a computable set under a total computable bijection is computable." Given the form of the theorem, there must be some example of a ...
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54 views

Recursion Tree method for solving Recurrences

I'm trying to find the tight upper and lower bounds for the following recurrence: T(n) = 2T(n/2) + 6T(n/3) + n^2, if n >= 3 = 1 if n <= 2 Drawing the ...
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41 views

recursive parsing parenthesis with explanation

I came across an explanation where (()())() gives you the sequence 0,1,2,1,2,1,0,1,0 using recursive parsing. Can someone give ...
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396 views

Pi's Recursiveness [duplicate]

I don't know if this will make sense, but: If $\pi$ is infinite and contains all strings of numbers including those of infinite length, then it must contain $\sqrt2$, and if $\sqrt2$ is infinite and ...
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Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?

So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as ...
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Recurrence for number of tile sequences of length n

We are placing tiles of colors red,blue and green in a row. Find a recurrence for number of tile sequences of length n (assuming unlimited supply is given) if a) No further conditions b) red tiles ...
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1answer
68 views

Problems understanding proof of s-m-n Theorem using Church-Turing thesis

I am reading Barry Cooper's Computability Theory and he states the following as the s-m-n theorem: Let $f:\mathbb{N}^2\mapsto\mathbb{N}$ be a (partial) recursive function. Then there exists a ...
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Recursion Tower of Hanoi [closed]

Consider the standard recursive solution to the Towers of Hanoi problem. In the traditional problem, all moves cost the same. Now, suppose the cost of a move is the size of the disk, with $1$ being ...
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Is there any procedure saying “this function is not obtainable without using recursion at least n times”?

It is known that $sum(x,y)=x+y$ is not obtainable from any compositions of basic functions $z,s,id^n_i$(zero, successor, projections) without using at least one recursion. also, $\times(x,y)=x\cdot y$ ...
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recursive function: non-recursive form possible?

Can the following recursive function be converted to a non-recursive form? $$f(x,c,\ell)=\frac{c-c^\ell}{1-c}+(c-2)\sum \limits_{k=1}^{\ell-1}f(x,c,k)$$ $$f(x,c,1)=c$$ $$c= \text{constant}$$ ...
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46 views

Solving a recurrence with 2 recurrences

I am trying to solve the following recurrence: $$T(n) = T\Big(\frac{n}{3}\Big) + T\Big(\frac{2n}{3}\Big) + O(n)$$ I do not want to use the Akra-Bazzi method nor draw out a recurrence tree. I do know ...
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Four people $(B,W,R,Y)$ are playing a game.

$4$ people, designated Black, White, Red, and Yellow, are playing a game. Each of them has unlimited number of balls, all of that person’s designated color. They work together to build a line of ...
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How to solve recurrence $T(n) = T(n/3) + T(2n/3) +n$ using Master Theorem

I'm trying to solve the following recurrence using Master Theorem, but I'm not used to seeing recurrences with to terms ( i.e. T(n)) for the cost of operations. I'm pretty sure that a should be 1 and ...
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Recursion involving Piecewise

Student $C$ tries to define a function $G$: $Z^{+}\rightarrow Z$ by the rule $G$($n$) = \begin{cases} \ 1, & \text{if $n$ is 1}\\ \ G(\frac{n}{2}), & \text{if $n$ is even} \\[2ex] G(3n-2), ...
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In recursion theory, is $\Sigma_{i=0}^y f(x,i,z)$ primitive recursive?

It is known that given ternary primitive recursive function $f$, the function $g$ defined as $g(x,y,z)=\Sigma_{i=0}^z f(x,y,i)$ is primitive recursive. I wonder if this formulation can be modified; ...
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39 views

Proving that $x_{n+1} = ax_{n}(1-x_{n})$ converges when $1<a<3$.

I need some guidance for proving $x_{n+1} = ax_{n}(1-x_{n})$ converges when $1<a<3$. What are some tips you guys could give me to help prove this point, without telling me how to do it? I'm ...
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1answer
32 views

Recursive Equations: Satisfiability and Defineability

I'm reading a functional programming book that comes from a very heavy mathematical perspective and I got stuck on trying to understand and expand a statement about recursive equations. It states: ...