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2
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3answers
162 views

finding explicit formula

The question ask us to guess an explicit formula for the sequence $$y_k = y_{k-1} + k^2 ,$$ for all integers $k$ greater than or equal to 2 and $y_1 = 1$ Can someone help me with this? so far what ...
1
vote
4answers
50 views

finding explicit formula through substitution method

The question ask us to guess an explicit formula for the sequence $$t_k = t_{k-1} + 3k + 1 ,$$ for all integers $k$ greater than or equal to 1 and $t_0 = 0$ Can someone help me with this? Because I ...
1
vote
2answers
30 views

Guess explicit formula using iteration

The question ask us to guess an explicit formula for the sequence $$s_k = s_{k-1} + 2k ,$$ for all integers $k$ greater than or equal to one and $s_0 = 3$ Can someone help me with this? Because I ...
1
vote
1answer
24 views

Solving a second level functional equation over all functions $g$

I am trying to find a closed form expression $f$ such that $$f(g(x+1) - g(x)) + f(g(x) - g(x-1)) = f(g(x))$$ For all functions $g$ I have concluded that for polynomials $$2^{n+1}f(0) = f(a_0 + ...
1
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2answers
59 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
0
votes
1answer
20 views

Check these recursive definitions for me?

Looking for Give a recursive definition of A) the set of odd positive integers B) the set of positive integer powers of 3 C) the set of polynomials with integer coefficients I have a. Basis: ...
2
votes
1answer
54 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
28 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
21 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
vote
0answers
30 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
78 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 ...
2
votes
2answers
132 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
0answers
44 views

Is the proof of non-totality of a function $f$ trivial if $f$ is recursively defined only on a subset of $\Bbb N$?

I define a fuction $*$ in a recursive way on a subset of natural numbers (with subset I mean that is $*$ define in the second argument only for $\Bbb N-\{0,1\}$ in other words we have $*:\Bbb N ...
5
votes
1answer
61 views

Compute a probability in Random Walk by Martingales

Let $X_n$ be the state at time $n$ of a Markov chain with these transition probabilities : $$p_{i,i+1}=p_i\qquad,\qquad p_{i,i-1}=q_i=1-p_i$$ $(a)$ Show that $Z_n=g(X_n)\,;\,n\geq0$, is a ...
1
vote
1answer
25 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
292 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
19 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
34 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
44 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: ...
1
vote
1answer
29 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
56 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 ...
1
vote
2answers
74 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
79 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
46 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
27 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
36 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?
-4
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
68 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
67 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 ...
1
vote
1answer
29 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
39 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 ...
3
votes
4answers
45 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
15 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 ...
0
votes
2answers
47 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 ...
1
vote
1answer
41 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 ...
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
105 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
25 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
22 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
30 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
76 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: ...
0
votes
1answer
28 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
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
85 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
47 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) = ...