# Tagged Questions

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 ...
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 ...
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 ...
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?
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 $O(\log N)$.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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 ...
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{ ...
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 ...
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