1
vote
1answer
18 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: ...
3
votes
4answers
39 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.
-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 ...
0
votes
1answer
18 views

How do you use induction on a recursive sequence using different variables?

I've been working on some recursive sequences for my Discrete class. I've understood most of them, but I've come to a new question which I'm confused about. A sequence $C_{0}$, $C_{1}$, $C_{2}$ is ...
1
vote
2answers
90 views

Convergence and limit of a recursive sequence

Let $p>0$ and suppose that the sequence $\{x_n\}$ is defined recursive as $$ x_1 = \sqrt{p}, \quad x_{n+1} = \sqrt{p + x_n}, $$ for all $n \in \mathbb{N}$. How can I show that $x_n$ converges, ...
1
vote
0answers
62 views

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: ...
0
votes
1answer
34 views

Find a formula for a sequence of the following character strings

Given strings: 0, 020, 0208020, 0208020180208020, 0208020180208020320208020180208020, ... I have to write a C++ program that computes the nth string. My problem is ...
1
vote
1answer
59 views

Recursive Square Root Futility Closet

This post on Futility Closet the other day: http://www.futilitycloset.com/2013/12/05/emptied-nest/ asked for the solution to this equation: \begin{equation}\sqrt{x+\sqrt{x+\sqrt{x...}}} = ...
1
vote
1answer
60 views

Let $x$ be a real number. To prove…

Let $x$ be a real number. Define the sequence $(x_n)_{n\ge1}$ recursively by $x_1=1$ and $x_{n+1}=x^n+nx_n$ for $n\ge1$. Prove that, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
0
votes
0answers
14 views

Recursion relation for the coefficients of the series solution around x=0 [duplicate]

ODE: the equation is the following; $$y''' + x^2y' + xy = 0$$ y=y0x0+y1x1+...+ynxn I have written equations for y, y', y'' and y''' there are: y = SUM y_k * x^k y' = SUM y_k * k * x^(k-1) y'' = ...
0
votes
0answers
58 views

How would you find a closed form expression for the following:

$$a_n = \frac{n}{n+1}a_{n-1} + \frac{n}{n-2}a_{n-2}\;,\;\;n\ge 3\; $$ For: $ 0 \leq n \leq 2 $, $a_n= n $
4
votes
2answers
79 views

Alternative unconditional form of $\sqrt{n -\sqrt{n -\sqrt{n -\cdots}}}$?

Consider $a_n$, where $$\begin{align} a_n &=\small{\sqrt{n -\!\!\!\sqrt{n -\!\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\sqrt{n - \cdots}}}}}}}}\end{align}$$ Using a recursive ...
0
votes
2answers
55 views

Equation of a curve whose difference in ordinate values form an arithmetic sequence

I have the following recurrence equation that I have obtained while trying to solve a problem:- $$T(0) = 1$$ $$T(n) = T(n-1) + 9n - 8: n \ge 1$$ The values of $T(n)$ for $n = 0,1,2,... $ are as ...
9
votes
5answers
497 views

Evaluating tetration to infinite heights (e.g., $2^{2^{2^{2^{.^{.^.}}}}}$)

The Problem How can you evaluate (i.e., get a value for) Tetration (i.e., iterated exponentiation) to infinite heights? For example, what would be the value of this expression? $$ ...
0
votes
2answers
175 views

How to derive a closed form of a simple recursion?

Consider: $$T(n) = 2 T(n-1) + 1$$ with $T(1)$ a positive integer constant $a$. I just stuck in finding a closed form for this simple recursion function. I would appreciate it, if someone gives me a ...
0
votes
1answer
78 views

Bounded recursive sequence

I would like to know if there are known bounded recursive sequence (non monotonic): It shouldn't be a constant, neither a convergent sequence, nor a periodic one. (I am not asking for a true random ...
4
votes
3answers
92 views

How to solve infinitely nested logarithms

I have an iterative process that starts with $$x_1 = \log_{10}(a)$$ Following iterations are as follows: $$x_2 = \log_{10}(a-b\cdot x_1)$$ $$x_3 = \log_{10}(a-b\cdot x_2)$$ $$x_4 = ...
0
votes
2answers
235 views

Give a recursive definition with initial condition…how did they get the answer?

The function $f (n) = 5n + 2, n = 1, 2, 3, \ldots $ Im sure its a simple problem, but im really confused...how did they get the answer ? could someone explain $f (n) = f (n - 1) + 5, f (1) = 7$ ...
2
votes
3answers
75 views

Closed form for $T(n) = n(T\left(\tfrac {n}{2}\right))^2$

I am trying to find a closed form for the following: $$T(n) = n(T\left(\tfrac {n}{2}\right))^2$$, with $T(1)=1/3$. I set $T(n)=T(2^m)=S(m)$ and then transformed the range of $S(m)$ to set ...
5
votes
2answers
249 views

A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of ...
3
votes
7answers
778 views

Find the nth term of a recursive sequence

I have a the following sequence: $$\begin{gather} a_1 = 3 \\ a_{n + 1} = 1 + \frac{a_n}{2} \end{gather} $$ How can I find the $a_n$ term?
1
vote
1answer
77 views

What kind of series / recursion is this?

I'm trying to find the explicit solution / sum of first n elements for the following sequence: d(2) = 2 d(n) = d(n/2) + n*log2(n) Can you help me to find out ...
1
vote
2answers
36 views

Simple linear recursion

$x_n=\frac{x_{n-1}}{a}+\frac{b}{a}$ with $a>1, b>0$ and $x_0>0$ I tried to solve it using the generating function but it does not work because of $\frac{b}{a}$, so may you have an idea.
0
votes
3answers
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!