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2
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1answer
61 views

Use principle of mathematical induction to show a function defined recursively is uniquely determined.

I'm having difficulty with the following taken from "Elementary Number Theory And Its Applications" by Rosen section 1.1 questions. "Use the Principal Of Mathematical Induction to show that the value ...
1
vote
1answer
29 views

Predicting eigenvalues of bigger matrices

Consider the following $(3 \times 3)$ matrix: $K_3 = \left( \begin{array}{ccc} a & -1 & 0 \\ -1 & a+1 & -1 \\ 0 & -1 & a \end{array} \right)$ The question has a quantum ...
1
vote
1answer
22 views

Resolve a recursive series

Consider the following recursive function: $f(n) = 1 + \sum_{i=0}^{n-1} f(i)$ with $f(0)=1$ I need to derive a non recursive form. By simply trying values, I have inferred that it must be ...
1
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0answers
34 views

Meaning of Biinterpretability.

I'm reading this paper: http://www.math.cornell.edu/~shore/papers/pdf/hyp9.pdf and I am struggling with the meaning of Biintereptability, to quote the paper A degree structure $D$ is ...
0
votes
0answers
9 views

A question concerning the domain of function that is recursively defined.

I have a problem concerning nested functions and whether they're well-defined. I have the following definition for all $t\in \mathbb{Z}$. A function $y_t:\mathbb{R}^3\rightarrow\mathbb{R}$ such that ...
1
vote
1answer
90 views

linear recursion $y_n=A \cdot y_{n-1}$

Let $a,b, \in \mathbb{R}$. Let $x_0=a, x_1=b$ and $x_n=\frac{x_{n-1}+x_{n-2}}{2}$ for $n \geq 2$ (i) Write the recursion in the form $y_n=A \cdot y_{n-1}$ where $A$ is a $2 \times 2$ matrix and ...
0
votes
4answers
1k views

Find a formula for a sequence of number

A sequence starts at $n=1$: $\{1, 4, 13, 40, 121, 364... \}$. Find an explicit formula that generates these numbers. Thanks a lot!
3
votes
1answer
3k views

Is 0^infinity indeterminate?

Is a constant raised to the power of infinity indeterminate? I am just curious. Say, for instance, is 0 raised to infinity indeterminate? Or is it only 1 raised to the infinity that is?
2
votes
2answers
141 views

Precisely, what is a primitive recursive definition?

If I understand correctly, the language of primitive recursive arithmetic has a distinguished function symbol $S$, together with a function symbol for each primitive recursive definition (hereafter ...
1
vote
1answer
28 views

Recursive sum of squares of prime numbers

Perfect squares always have recursive sum of their digits(e.g. $361\rightarrow 10 \rightarrow 1$) as either $1, 4, 7$ or $9$. But if the perfect square is square of a prime number, except of $3$ . ...
2
votes
3answers
336 views

recursive digit sum of cubes of integers.

For every cube of an integer, the recursive sum of its digits , e.g. 729 -> 18 -> 9 etc. is always 1,8 or 9. With a computer program i checked this phenomenon up to 1000000000. In my prior ...
-1
votes
2answers
46 views

Define this recursively : $f(n) = 3n - 4$

Define this recursively : $f(n) = 3n - 4$ I thought this is how you recursively define a function? $f^{-1}(n): y = 3n-4$ $y+4 = 3n$ $f^{-1}(n) = (y+4/3)$ But the answer is $f(n) = f(n-1) + B$ ...
0
votes
0answers
31 views

Amount of sub-matrices created by Laplace expansion

I have created a program that solves a matrices determinant using the Laplace expansion method, and I was wondering if there is a equation which provides how many sub-matrices are created and used in ...
2
votes
0answers
51 views

Dynamic Programming: Stock Exercise

I'm having a trouble dealing with this problem: Since future market prices, and the effect of large sales on these prices, are very hard to predict, brokerage firms use models of the market to ...
2
votes
2answers
53 views

Master theorem - why the log factor?

I think I finally managed to fully understand the master theorem but there's one thing left in the second clause (I'm following here: ...
3
votes
2answers
137 views

How do we know that every halting Turing Machine can be expressed as a recursive function?

I've hear many times that a major result in Recursion Theory is the equivalence of Turing and Godel's models: the functions implementable on a Turing machine are precisely the functions that can be ...
1
vote
1answer
31 views

Coloring a binary tree

Working through a problems practice coloring, I have found a problem that has me stumped. The problem states: For $n \in \mathbb{R}_{>o}$ the binary tree is defined recursively as follows. The ...
0
votes
1answer
59 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 ...
6
votes
3answers
198 views

Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
1
vote
2answers
88 views

Use the recursive definition of summation together with mathematical induction to prove a sequence

Use the recursive definition of summation together with mathematical induction to prove that for all positive integers $n$ if $a_1, a_2,\ldots, a_n$ are real numbers, then $$\sum_{k=1}^n(3a_k - 2k + ...
1
vote
2answers
109 views

Find an explicit formula for the recursive sequence (tips?)

Problem: A sequence is defined recursively as follows: Sk = 2k - Sk - 1, for all integers k greater than or equal to 1 S0 = 1 Use iteration to guess the explicit formula for the sequence. Use ...
3
votes
3answers
50 views

Prove convergence and find the value of the limit of the sequence

Sequence $$a_{n+1}=(1+\frac{1}{3^n})a_n$$ $$a_1=1$$ The question asks to prove its convergence and find its limit. I have tried all the usual ways but am unable to solve it. The question also says ...
0
votes
1answer
28 views

Diagonalization out of partial recursive functions

So generally partial recursive functions don't diagonalize. But isn't this function an exception? $\phi(x)=\lambda_{x}(x)+1 $ if $\lambda_{x}(x)$ halts and $0$ else. Completely no clue... It seems ...
1
vote
0answers
22 views

Show $f(x)$ is partial recursive using the

So I want to show that the function $f(x)=0 $ if $x$ is even and not defined otherwise, is partial recursive using the $\mu$ operator (bounded search function, which is partial recursive). Plan is to ...
0
votes
4answers
37 views

Given n flips, what is the expected number of times the sequence HT shows up?

Suppose a fair coin is flipped $n$ times - what is the expected number of appearances of the sequence HT? I know it's $\sqrt n$, can anyone provide an explanation why this is true?
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votes
1answer
38 views

Find a recursive algorithm to find $a^{2^n}$

Edit1: Used Latex. =] Edit2: Thanks for the guidance to the users below. Really helped me out editing the post and guidance on the math problem. The question gave me a hint: $a^{2^{n+1}} = (a^{2^n}) ...
1
vote
2answers
72 views

Definition by Recursion: why is the existence part not (almost) obvious?

I saw the following statement. Let $H$ be a set, let $e\in H$ and let $k:H\rightarrow H$ be a function. Then there is a unique function $f:\mathbb{N}\rightarrow H$ such that $f(1)=e$, and that ...
0
votes
0answers
68 views

Kleene's recursion theorem

Would anybody be able to provide me (someone with little familiarity with the subject matter) with a bit of background to the Recursion Theorem and guide me towards some texts on its mathematical and ...
4
votes
0answers
157 views

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 ...
1
vote
1answer
31 views

How to find a pattern in this recursive sequence algorithmically?

I'm trying to find the closed-form of a sequence algorithmically. Here is the recursive sequence: $$w_k=w_{k-2}+k, \forall k \in \Bbb{Z} | k \geq 3, w_1=1, w_2=2$$ which produces this sequence: ...
2
votes
2answers
50 views

Recurrence relation practice problem that I can't figure out

Thanks for taking the time to look at this problem. I'm trying to prepare for a test on Monday by doing some extra odd numbered problems from my textbook. I'm having a lot of trouble trying to solve ...
1
vote
3answers
160 views

Recurrence relations (Big-O notation)

Say I'm given a recursive function such as: function(n) { if (n <= 1) return; int i; for(i = 0; i < n; i++) { function(0.8n) } } ...
3
votes
4answers
47 views

Is this a correct recursive sequence definition?

Take this definition: Is this definition of $s_k$ for $k\ge2$ correct? $s_k=6a_{k-1}-5a_{k-2}$ but where does the $a$ term come from? The book swaps $a$ and $s$ interchangeably.
0
votes
2answers
48 views

Recurrence Relationship Questions

Consider the recurrence defined by: $$G_0 = 0\\ G_n = G_{n-1} + 2n - 1$$ Determine what Gn is for several values of n to determine a formula for Gn. $2n$ $n$ $2n-1$ $n^2$ *I believe this one is ...
0
votes
2answers
16 views

Recursion, Explicit Equasion

Prove $\ a_{n}<2^{n} $ for every natural number n, where $\ a_{n} $ is defined recursively by $$ a_{1}=1, a_{2}=2, a_{3}=3, a_{n}=a_{n-3}+a_{n-2}+a_{n-1},\ for\ n>=4$$ Once I get the explicit ...
1
vote
1answer
45 views

Simple Recurrence Questions [closed]

$r$ is a real number, define a recurrence relationship for $A_n$ $$A_0 = 1\\ A_n = r\cdot A_{n-1}$$ Question: What is the value of $A_4$ $4(A{n-1})$ $r^4$ $1$ $4r$ I've pretty much eliminated ...
0
votes
1answer
29 views

Recursively defined sequences

So, this question has been giving me a little bit of trouble. It's supposed to be just a few lines, and I know that I don't need to write out the base case, recursive step, and restriction. I ...
1
vote
0answers
32 views

A Problem about Recursion

Consider recursion $$a_n+c_1a_{n-1}+\cdots+c_ma_{n-m}=0~~~~~~~~~~(1)$$ Let $\lambda^m+c_1\lambda^{m-1}+\cdots+c_m=0$ be the characteristic function and $(\lambda_1,n_1),\cdots,(\lambda_s,n_s)$ be ...
2
votes
1answer
126 views

Guess an explicit formula for recursively defined sequence

Was given this as a question. "Use iteration to guess an explicit formula for the recursively defined sequence and then prove that the formula is a solution to the recurrence using induction: ...
0
votes
1answer
27 views

Prove uniqueness of recursive function

I am currently reading Cutland's Computability and would like to figure out how to solve Theorem 4.2 which states: Let $x=(x_1 \dotsc x_n)$, and suppose that $f(x)$ and $g(x,y,z)$ are functions; ...
0
votes
1answer
24 views

how to write a recursive definition

so the question asks to define s(n) as the number of strings of a's b's and c's of length n that do not contains "aa". write a recursive definition for s(n). what is s(0),s(1),s(2),s(3). i had to ...
0
votes
1answer
31 views

How to deduce from a recursive variant of the triangular numbers the non-recursive form?

This is probably simple (if not, my apologies beforehand), but I have the following formula: $$f(0) = 2$$ $$f(x) = 6x + 2 + f(x -1) \; \text{ where } x \in \mathbb{N}_{\gt 0}$$ The formula actually ...
2
votes
0answers
86 views

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 ...
0
votes
1answer
25 views

proof using a recursive definition

I am doing a 2-part question. Thus far, I have finished the first part, requiring me to make a recursive definition of a set "S" of all binary strings, starting with a 1. I have: Base: 1 Recursion: ...
-1
votes
1answer
49 views

Proof of a sequence with recursion

The problem asks to prove the following to be true. $$F^2_{n+1} - F_{n+1} F_n - F_n^2 = (-1)^n$$ Anyway, I've tried looking at this or similar proofs for going on an hour now, pretty much the only ...
1
vote
2answers
23 views

Question over recursive definitions

Let $E$ be the set of even integers. Then the Base Case: $0\in E$ And the constructor case would be If $n \in E$ then so are $n+2$ and $-n$. This makes sense. But would the Base case and constructor ...
2
votes
5answers
91 views

How to find first five terms of sequence?

I'm new to recursion so please bear with me. I have to find the first five terms of a sequence with initial conditions $u_1 = 1$ and $u_2 = 5$, and, for $n \geq 3$, $$u_n = 5u_{n−1} − 6u_{n−2}.$$ I ...
3
votes
2answers
49 views

$T(n)=T(cn) + T((1-c)n)+1$ while $0<c<1$

Question: $T(n)=T(cn) + T((1-c)n)+1$ $0<c<1$ and $T(1)$ is constant. My thoughts: I'm trying to solve this recursion using Induction, but I think I got it all wrong. My guess is that $T(n) = ...
3
votes
1answer
128 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
0
votes
1answer
43 views

Discrete Math Recursive Functions for Strings

Recursive Functions for Strings Construct a recursive definition for the following string function over the alphabet {x,y}: f(x) returns the string where every x is replaced by xy and every y is ...