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Questions tagged [recursion]

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 a function for some inputs in terms of the values of the same function on other inputs. Please use the tag 'computability' instead for questions about "recursive functions" in computability theory

Recursion is the process of repeating items in a self-similar way. The most common application of recursion is in mathematics and computer science, in which it refers to a method of defining functions in which the function being defined is applied within its own definition.

Basically, a class of objects exhibiting recursive behaviour can be characterised by two features:

  • There must be a base criterion for which the function should not call itself.

  • Every other iteration of the function should move it closer to the base condition.

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Recurrence relation for the colored balls problem

Right now I am attempting to solve the recurrence thats solves a famous problem (the colored balls problem: https://mathoverflow.net/questions/41939/a-balls-and-colours-problem) given by $2i(n-i)Z_i = n(n-1) + (n-i)(i+1)Z_{i+1} +…
user19164
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Recurrence relations

For each integer $n ≥ 1$, let $t_n$ be the number of strings of $n$ letters that can be produced by concatenating (running together) copies of the strings $“a”$, $“bc”$ and $“cb”$. For example, $t_1$ = $1$ (“a” is the only possible string) and $t_2$…
dave
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Formula for a recursive function

Given the recursive function $T: \mathbb{N}_0 \to \mathbb{R}_+$ $$T(n) ≤ max_{1 ≤ k ≤ n - 1}\{T(k-1) + T(n - k) + c n^2\}$$ with c > 0 constant, I want to determine an absolute formula to quickly determine a supremum or an upper limit as low as…
moran
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Proof that mappings $K$ and $L$ are primitive recursive.

Let $J$ be the function: \begin{equation*} J(m,n)= \begin{cases} n^2+m \text { if } m\le n \\ m^2 + m + (m-n) \text { if } m > n \\ \end{cases} \end{equation*} Let $K, L$ such that $K(k)$ is the unique $n \in \mathbb{N}$ for which there is some…
Sara
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$2^n=na_n+na_{n-1}-a_{n-1}$ by range transformation

I want to range transform $2^n=na_n+na_{n-1}-a_{n-1}$ to get rid of the $2^n$ term and then solve it with any other method (seems like telescoping will work once it's reduced). I've tried transforming it by $b_n=a_n/2^n$ which…
Whitesizzle
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Demonstrating Strassen's method using domain transformation: $T(n)=7T(n/2)+an^2$

I want to solve the recurrence for Strassen's method (for multiplying square matrices) with domain transformation and get a closed form. The equation is given below: $T(n)=b$, at $n=2$ $T(n)=7T(n/2)+an^2$, at $n>2$ I know that since I have to cancel…
Whitesizzle
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Solving $\scriptsize a_n=\sqrt{a_{n-1}+\sqrt{a_{n-2}+\sqrt{a_{n-3}+\ldots}}}$ with range transformation

This is a practice problem provided by a textbook on recurrences. Solve using range transformation: $a_n=\sqrt{a_{n-1}+\sqrt{a_{n-2}+\sqrt{a_{n-3}+...}}}$, where $a_0$ =4 The hint is to view the first few terms to find a pattern. So what I've got:…
Whitesizzle
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Solving $a_n=5a(n/3)-6a(n/9)+2log_3n$ using domain transformation

$a_n=5a(n/3)-6a(n/9)+2log_3n$, For $n\ge9$ and n is a power of 3. $a_3=1$, and $a_1=0$ Transforming the first two terms is straightforward, but I'm not sure what to do with the log term. Should I rewrite it somehow? The fact that it isn't attached…
Whitesizzle
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Is a linear random walk with jump recurrent?

Let $\lambda_0=10^5$ or any other large integer. Define the recursive "process": $\lambda_t=\text{sample from a Poisson distribution with mean }\lambda_{t-1}$. Is this process recurrent? I mean, after a long time, does the probability of reaching…
user3813057
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How to state a recurrence that expresses the worst case for good pivots?

The Problem Consider the randomized quicksort algorithm which has expected worst case running time of $\theta(nlogn)$ . With probability $\frac12$ the pivot selected will be between $\frac{n}{4}$ and $\frac{3n}{4}$(a good pivot). Also with…
committedandroider
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Explicit formula for recursive geometric/arithmatic series

In my Algebra 2 class, we have come upon a question that the class could not solve, and that the teacher has neglected to remove from the given packet for several years because of this. The problem is fairly easy to derive a tentative recursive…
Pip
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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 please solve it and explain to me what's going on at…
walter
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Runtime of recursive algorithm - Master's Theorem

I wrote a computer program that solves a question, and I am interested in knowing what is the runtime. My aim is for $O(\log n)$, and I'd like someone more experienced (and smarter?) to review my calculations and analysis The algorithm is very…
Oria Gruber
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function with a recurrence relation

I have this recursive equation: $$\begin{align*} F(m,n)&=F(m,n-1)+F(m-1,n)-F(m-1,n-1-m)\\\\ F(m,0)&=F\left(m,\frac12m(m+1)\right)=1\\ F(m,i)&=0\text{ if }i<0\text{ or }i>\frac12m(m+1) \end{align*}$$ is there a way to solve this recursive equation to…
Motaz Hammouda
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Stability with 2 dimensional recursion functions.

First of all, hello. I'm having trouble determining whether fixed points are stable or unstable. I have a recursion function:…
BlockCat
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