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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$\mu-$recursive functions

In my book there is the following: Although the class of primitive recursive functions contains a great many functions of practical interest, it does not include all the Turing-computable or ...
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Undecidable definition of pure function

I am trying to come up with a formal definition for functional purity in a simple programming language (think JavaScript). What I've got so far is this: DEFINITION: A statement is impure if ...
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Finding a recursion

I am supposed to find a recursion for the following sequence: $$a_{n} = (1+\sqrt{s})^{n} + (1-\sqrt{s})^{n}$$ where $s \in \mathbb{N}$ fixed. I tried playing around with it using the binomial theorem, ...
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30 views

Right notation to recurse over a sequence or list

I have a function $f(x, a)$ which is invoked over all the elements of a sequence feeding the result to the next call, with $x$ being the next element in the list and $a$ the accumulated result. What ...
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Why is $y + 1$ infinite?

This is related to SO question : http://stackoverflow.com/questions/30150877/why-does-this-cause-ghci-to-hang but I'm having difficulty understanding why Haskell enters an infinite loop but since ...
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Discrete math, Showing a recursive equation as equivalent to a non recursive equation.

I'm having trouble with this: Show that this recursive function: $L(n) = \{0 : n = 1\ ,\ \lfloor(L(n/2))\rfloor +1 : n \gt 1\}$ is equivalent to this non-recursive equation: $L(n) = ...
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29 views

Where does the root of this tree come from?

I am doing a practice question from Midterm Dynamic Programming The Problem : Consider a row of n numbers a1, ..., an. The numbers are all positive, and n is even. We play a game against an ...
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45 views

How to apply Master theorem to this relation?

This is the definition of master theorem I am using(from Master Theorem) I am trying to use that master theorem to find the tight bound for this relation $T(n) = 9T(\frac{n}{3}) + n^3*log_2(n)$ ...
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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, ...
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Indexing for partitions

I'm using spatial hashing for broad-phase collision detection and I'm trying to squeeze some more performance out of it. Currently, it creates a new hashset for every partition which works well ...
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Differential Equations: Recursive Functions

Functions I have some familiarity with look so, $y^\prime(x) = \tan(x+2)$: straightforward expression of the first derivative of y as a function of x. But say I have a function, $y^\prime(x) = ...
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Proving the principle of definition by generalized recursion using the inductive closure of an induction system

I'm working through Hinman's Fundamentals of Mathematical Logic in order to review some things, and got stuck in an exercise from section 1.2. Specifically, he asks us to prove (what he calls) the ...
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47 views

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 ...
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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: http://mathoverflow.net/questions/41939/a-balls-and-colours-problem) given by $2i(n-i)Z_i = ...
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32 views

Solving recurrence with moebius inversion

what's up folks? I'm solving the red book of math problems, problem 16 which is to solve the following recurrence relation: $\sum_{k=1}^n {n \choose k} a(k) = \frac{n}{n+1}$ PS: ${n \choose k} = ...
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32 views

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, $t_1$ ...
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11 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 ...
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24 views

$2^n=na_n+na_{n-1}-a_{n-1}$ 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 ...
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Proving $1+2CZ+3C^2Z^2+…=1/(1-CZ)^2$, considering $\sum\limits_{i=1}^{\infty}c^iZ^i=(1-CZ)(1+2CZ+3C^2Z^2+…)$

I'm told that we can prove this common identity for solving generating functions: $1+2CZ+3C^2Z^2+....=1/(1-CZ)^2$ Using only the property ...
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22 views

Why is $X_m$ and $Y_m$ not included in the shaded region(where median can lie)?

This problem is from Algorithms, problem 2 The Problem Given two sorted list of numbers $X$[1..$n$] and $Y$[1..n]. we need to come up with a O($\log n$) time algorithm to find the median of the 2$n$ ...
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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 $n=2$ ...
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29 views

Getting rid of exponents with n when solving with annihilators: $a_n=a_{n-1}+2a_{n-2}+2^n+n^2$

To solve the following with annihilators: $a_n=a_{n-1}+2a_{n-2}+2^n+n^2$, for $n\ge2$, with initial conditions $a_1=0$ and $a_0=0$ we would have to get rid of the $2^n$ term at least, otherwise any ...
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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 ...
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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 ...
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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 ...
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51 views

How do I make my TI-89 evaluate a recursive function?

On my TI-89 I can assign variables recursively such as: $1\to x$ returns 1 $x+1 \to x$ returns 2 $x+1 \to x$ returns 3 etc. How could I do functions the same way: $x \to f(x)$ returns Done $2\cdot ...
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58 views

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$ whenever $n$ is a positive integer. Basis step: $P(1)$ is true because $1 \cdot 1!=(1+1)!-1$ evaluate to $1$ on both sides. Inductive ...
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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 ...
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Manipulating series to find the recursive formula

Ok so I am stuck. I need to get all the $n$'s to $=0$ but I can't reduce my series which has $n=2$ to $0$ because then I will have undone all my work in the first place to get all the $X^n$'s to the ...
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Does this recurrence relation run in $ \Theta(n) $?

This is the recurrence relation I am trying to solve: \begin{align} T(n) & = 2 \cdot T \left( \frac{n}{4} \right) + 16, \\ T(1) & = c. \end{align} I broke this down (i.e., solved this ...
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Asymptotic behavior a recursion involving min/max

Usually when I face solving recursions I use generating functions but I'm not aware of any "tools" to use when min/max expressions are involved. For example, I have the following recursive term: ...
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How to show that recurrence $T(n) \in \Omega(n^{0.5})$ using proof by induction?

This is recurrence $T(n)$ $ T(n) = \begin{cases} c, & \text{if $n$ is 1} \\ 2T(\lfloor(n/4)\rfloor) + 16, & \text{if $n$ is > 1} \end{cases}$ This is my attempt to show that $T(n) \in ...
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Binary strings and recurrence relations [closed]

So the problem is: How many binary strings of length n contains 111? Give a recurrence relation Tn, where Tn is the number of binary strings of length n that contains 111. How could we possibly ...
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functions and recursions

The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4. If s(n) is rewritten in the form s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms ...
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Chaos theory and fractal geometry: Constructing from data

I understand that fractal geometry represents behaviour of 'chaotic' system, if I am not wrong. And also, fractals are generated by a recursive function. But, lets say I have random data lying with ...
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52 views

Discrete Mathematics Fibonacci Sequence

I am studying for the final exam in my Discrete Mathematics class and came upon the following problem on the study guide we were given. Given the following algorithm: If $n = 0$, then $f(n) = 0$ ...
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55 views

Proof of Newton Girard formula symmetric polynomials

Newton Girard formula states that for $k>2$: \begin{equation} p_k=p_{k-1}e_1-p_{k-2}e_2+\cdots +(-1)^{k}p_1e_{k-1}+(-1)^{k+1}ke_{k} \end{equation} where $e_i$ are elementary symmetric functions and ...
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93 views

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 ...
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uncountable repetitions

I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence: Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be ...
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Mini Tetris Winning Configuration

So here's the problem: A winning configuration in the game of Mini-Tetris is a complete tiling of a 2 x n board using only the three shapes shown in Figure 1. By allowing rotations, there can be ...
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Finding an explicit formula for An in terms of n.

Let $A_{1}= 5, A_{2} = 7,$ $A_{n+1} = A_{n} + 6A_{n-1}$. Find an explicit formula for $A_n$ in terms of $n$. Suppose $p_{n}$ is a sequence defined by $p_{1} = 0, p_{2} = 1$ & $\forall n$, ...
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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 ...
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Is every natural recursive relation necessarily holomorphic?

Define the set of algebraic primitive recursive relations as the set of functions defined by: $$ F(n,a,k) = F(n-1,F(n-1,F(n-1,a,a),a)...,a)_{\text{nested to depth k}}$$ $$ F(0,a,k) = a + k $$ Along ...
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42 views

Expressing a function's value using finite differences

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and let $x = (x_0, x_1, x_2, \dots)$ be a sequence of pairwise distinct real numbers. For every $n \in \{1, 2, \dots\}$ and every ordered $(n+1)$-tuple ...
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The Ackermann's function “grows faster” than any primitive recursive function

I am looking at the proof that the Ackermann's function is not primitive recursive. At the part: "We will prove that Ackermann's function is not primitive recursive by showing that it "grows ...
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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 ...
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30 views

Working out recursive functions

I know that a function is defined recursively when it calls itself to progressively converge on a solution based on an initial condition. For example if we consider the function defined by $$Sq(1) = ...
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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 ...
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20 views

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: \begin{align*} f_{\alpha,\gamma} \left( \begin{array}{c} t\\ v ...
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166 views

Complicated Multivariate Recurrence Relations For Generating Polynomials

I have the following multivariate recurrence relations all from the same system: First, suppose that $0\le k\le j\le m$, and let $N$ be an independent integer. Then we have for expressions $a(k,~ m,~ ...