Tagged Questions
2
votes
1answer
42 views
Number of Distinct Resistances that can be produced from n equal resistance resisters
Here is an interesting problem:
The number of distince resistances that can be produced from n equal resistance resisters is given below.
The Sequence
Surprisingly this is also equal to the number ...
28
votes
2answers
536 views
Predicting Real Numbers
Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
1
vote
2answers
50 views
Lengths of increasing/decreasing subsequences of a finite sequence of real numbers
Let $x_1,\ldots,x_n$ be a finite sequence of real numbers. Let $f(\{x_i\}_{i=1}^n)=f(\{x_i\})$ be the length of the largest non-decreasing subsequence, and let $g(\{x_i\})$ be the length of the ...
0
votes
0answers
22 views
Given an X and a Y, how to find the equation
I've just been curious lately. If you have an X for lets say 0-50 and corresponding Y values, is there a way to determine the equation without just guessing and checking and trying to find a pattern?
2
votes
1answer
91 views
How to calculate $ 1^k+2^k+3^k+\cdots+N^k $ with given values of $N$ and $k$? [duplicate]
Here $ 1<N<10^9$ and $0<k<50$
So we have to calculate it in order of $O(\log N)$.
0
votes
1answer
54 views
Sequence Question from past post
I recently saw a post about sequences. This made me remember some other post someone had posted here on Math.SE. He did not want answers but wanted general ways to tackle them. I did spend an hour or ...
4
votes
5answers
179 views
What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$,$F(1)=b$ and $a,b>0$?
What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$, $F(1)=b$ and $a,b>0$ ? It seems to be simple generalization of Fibonacci sequence but I can't find closed form for ...
10
votes
2answers
140 views
Integer sequences which quickly become unimaginably large, then shrink down to “normal” size again?
There are a number of integer sequences which are known to have a few "ordinary" size values, and then to suddenly grow at unbelievably fast rates. The TREE sequence is one of these sequences, which ...
2
votes
1answer
61 views
Question on pathological sine function
Some years ago I came across what was defined as "pathological" function defined as:
$$
f(x)=\sum_{k=1}^\infty \frac{1}{k^2}\cdot \sin\left(k!\cdot x\right)
$$
It was mentioned (in an article I ...
8
votes
1answer
156 views
Do roots of a polynomial with coefficients from a Collatz sequence all fall in a disk of radius 1.5?
Consider a modified version of Collatz sequence:
$C(n)=\left\{ \begin{array}{ll} \frac{3n+1}{2} & n\ \mathrm{odd} \\ \frac{n}{2}& n\ \mathrm{even}\end{array} \right.$
Let $F_n$ be the ...
0
votes
1answer
61 views
Gradually rising or falling numbers
I'm looking for a number series I can use for gradually rising or falling numbers. The number series should not be linear and should converge to a number at some point.
(Sorry I'm really scared of ...
3
votes
3answers
191 views
Convergence of $x_n = \cos (x_{n-1})$
I define the sequence $x_n = \cos (x_{n-1}), \forall n > 0$.
For which starting value of $x_0 \in \mathbb{R}$ does the sequence converge?
2
votes
1answer
121 views
Number of combinations of $k$ numbers using arithmetic operations
What is the maximum number of positive rational values that can be obtained by combining $k$ positive integers using only addition, subtraction, multiplication, division, and parentheses? Assume that ...
7
votes
0answers
164 views
Largest $x$ such that the power tower (tetration) $x^{x^{x^{x^{…}}}}$ converges? [duplicate]
Possible Duplicate:
Infinite tetration, convergence radius
Recently in this thread, Pseudo Proofs that are intuitively reasonable, I learned that
...
2
votes
2answers
217 views
A sequence of nested fractions with a counter-intuitive limit
Given $a,b\in\mathbb C$, let us construct the following sequence:
$$\begin{align}
a+b&=a+b\\
\cfrac a{a+b}+\cfrac b{a+b}&=1\\
\cfrac a{\cfrac a{a+b}+\cfrac b{a+b}}+\cfrac b{\cfrac ...
5
votes
0answers
150 views
$2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$
I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$.
Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
2
votes
3answers
199 views
A binet-like formula for $\small 1 , 2 , 3 , 6 , 9 , 18 , 27, 54, 81, \ldots $?
This is more a "recreational" problem. By another question I came to the question for a closed-form-formula for this sequence $\small 1 , 2 , 3 , 6 , 9 , 18 , 27, \ldots $ which is just the mixture of ...
