It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The …

There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested in …

While playing with something totally irrelevant I stumbled upon the recurrence: $$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$ It turns out that given $a_0 = 1, a_1 = 1$, $$lim \frac{a_{2 …

Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$. I denote the polynomials $p_l(t) = …

As in A072868 described by OEIS; Defined by $\sigma(\sigma(n)-n) = n$. Since these numbers are important in regard to many things, specially mersenne primes, since ${n-1 \over 2} …

To construct J. H. Conway's look-and-say sequence, begin by putting down a 1 as the first entry. The other entries are found by saying the previous entry aloud, and writing what yo …

There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accid …

Let $p_n$ be the nth prime and $p_L$ be closest to its square root: \begin{equation} p_L^2 \approx p_n \approx x \end{equation} Let $\sigma \in Z^+$ be a positive integer consta …

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulat …

Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post. For integer $n\ge0$, let $s(n)$ denote the sum of the digits …

I have to find the expression of $(y_n)$ defined by : $$y_{n+1}=a y_n+b z_n+c$$ where $(z_n)$ is an arithmetico-geometric sequence : $$z_{n+1}=d z_n+e$$ and $a,b,c,d,e$ real nu …

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric: If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent …

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$. By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this be …

Some time ago, while putting my nose in the Sloane's Online Encyclopedia of Integer Sequences, I came over the sequence A019568 defined as follows: $a(n):=$ the smallest positi …

Hello! I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own. Suppose I am drawing k numbers out of n numbers labeled from 1 to n. Consideri …