# Tagged Questions

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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### 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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### Conway's Game OF Life maximum periods on a set x by x game board.

I have taken interest in Conway's Game of Life and want to know if you guys can help me with a mathematical problem :) That is what this website is for right? You need to be familiar with the rules ...
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### Computing cube roots using three number means

I've asked a question some time ago about Computing square roots with arithmetic-harmonic mean but it turned out that this method is exactly the same as Newton's method (or Babylonian method) for ...
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### Skiponacci: $p | a_p$ Alternate Solution

For the Skiponacci sequence: $a_0=3, a_1=0, a_2=2,$ and $a_{n+1}=a_{n-1}-a_{n-2}$ for $n>2$, prove that any prime $p$ divides $a_p$. Is there any alternate solution other than using ...
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### Distribution of $\max_{n \ge 0} S_n$, random walk.

Say I have a random walk that's a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...
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### uniform convergence in recursive function

Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to \mathbb{R}$ an increasing function (...
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### How to get the period of oeis.org/A130166 other than by trail?

oeis.org/A130166 a(0)=1; a(n)=prime(mod(a(n-1),1000)) starts with: ...
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### Induction on Primitive Recursive Function

The set $F_{n}$ of primitive recursive function symbols of arity $n$ can be defined inductively as: \begin{align} & Z, \text{Succ} \in F_{1} & \\ &\pi_{j}^{n} \in F_{n} \quad \text{for ...
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### 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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### What is the maximum amount of solutions to $f(x+1)f(x)= ax^2+bx+c$.

$f(x)$ is going to be in the form $mx+h$ thus, $(mx+m+h)(mx+h) = ax^2+bx+c$. With basic algebra $m= \pm \sqrt{a}$. Also $(m+h)(h)=c$. I would guess that because $(m+h)h=c$ has two solutions max if $m$ ...
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### Recursion asymptotic growth function proof of complexity

I have the following recursive function(an example from a textbook): $$T(n)=\begin{cases}1&n=1\\T(\lfloor\tfrac n2\rfloor)+T(\lceil\tfrac n2\rceil)+1&n>1\end{cases}$$ A recursion is ...
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### Ackermann function is not primitive recursive

The function of the Ackermann function is defined as $$A_{0}(y)= y+1$$ $$A_{x+1}(0)= A_{x}(1)$$ $$A_{x+1}(y +1)= A_{x}(A_{x+1}(y))$$ I want to show that the function of ackermann is primitive ...
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### What is the solution to this recursion?

Take $a_0=10^6$. What is $a_n$ (asymptotically) where $a_{i+1}=a_i+\sqrt[\alpha]{a_i}$ where $\alpha>1$? How fast does $a_n$ grow?
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### recursion trees and big theta bounds

Draw recursion trees and use them to find big theta bounds on the solutions to the following recurrences. For each, assume that T(1) = 1 and that n is a power of the appropriate integer. ex) T(n) = 8T(...
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### Proving that for $F_k = F_{k-1} + F_{k-2}$, $F_k$ is even iff $3|k$

Consider the recursion $F_k = F_{k-1} + F_{k-2},$ $k\geq 2,$ $F_0 = 0$ and $F_1 = 1$, then show that $F_k$ is even iff $3|k$ I tried to do a proof by induction: The statement is true for base ...
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### Solving systems of reccurence equations to get number of recursions to reach a stationarity point?

I'm tring to solve a system of recurrence equations to then formalize a formula depending on the number of recursions to calculate this number of recursions to reach the stationarity of the system ...
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### Primitive recursivness of a function. How does the function work?

So, I need some help with an homework assignment. Firstly: understanding the following function: $h(x) = \prod_{m=0}^{f(x)} m*f(m)$ From my limited knowledge of the product of sequences my guess is ...
Right now I am attempting to solve the recurrence thats solves a famous problem (the colored balls problem: http://mathoverflow.net/questions/41939/a-balls-and-colours-problem) given by $2i(n-i)Z_i = ... 0answers 30 views ### Formula for a recursive function Given the recursive function$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 ... 0answers 33 views ### 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 ... 0answers 45 views ### 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$...
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 ...