Questions tagged [sequences-and-series]

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On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence: It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
Sebastien Palcoux's user avatar
32 votes
0 answers
2k views

The easily bored sequence

If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
Alessandro Della Corte's user avatar
31 votes
0 answers
2k views

A question related to the Hofstadter–Conway \$10000 sequence

The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
Alkan's user avatar
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28 votes
0 answers
695 views

Does this infinite primes snake-product converge?

This re-asks a question I posed on MSE: Q. Does this infinite product converge? $$ \frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
Joseph O'Rourke's user avatar
23 votes
0 answers
1k views

Is A276175 integer-only?

The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
uvdose's user avatar
  • 593
19 votes
0 answers
770 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is $$\...
Henry's user avatar
  • 830
18 votes
0 answers
746 views

Two curious series for $1/\pi$

On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have $$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$ where $(\frac{\cdot}p)$ ...
Zhi-Wei Sun's user avatar
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16 votes
0 answers
392 views

Division of a square and value of a disk

[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk I cam across this problem ...
user33624's user avatar
  • 477
16 votes
0 answers
769 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
Stefan Kohl's user avatar
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16 votes
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1k views

Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
user avatar
14 votes
0 answers
367 views

Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?

On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
Dan's user avatar
  • 2,341
13 votes
0 answers
305 views

Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$

This is inspired by this Math.SE question, for $a=1$. Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series $$\sum_{n=1}^\infty \frac{(2+\...
Clement C.'s user avatar
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11 votes
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319 views

Does any real function have a Lipschitzian restriction on $D$?

Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
Dattier's user avatar
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10 votes
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493 views

New series for $\zeta(5)$ involving second-order harmonic numbers

In 1997 T. Amdeberhan and D. Zeilberger proved that $$\sum_{k=1}^\infty\frac{(-1)^k(205k^2-160k+32)}{k^5\binom{2k}k^5}=-2\zeta(3).\tag{1}$$ In 2008 J. Guillera obtained that $$\sum_{k=1}^\infty\frac{(...
Zhi-Wei Sun's user avatar
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10 votes
0 answers
662 views

Is there a proof that OEIS-A002387 is $[ e^{n-\gamma} ]$?

Based on the comments on OEIS-A002387: $a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,... it is likely, that the sequence $a_{n}$ coincides with $[ e^{n-\gamma} ]$ , where $\gamma$ is the Euler-Mascheroni ...
user avatar
9 votes
0 answers
318 views

When exactly is the principal AGM equal to the optimal AGM?

Definitions Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
Wane's user avatar
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9 votes
0 answers
389 views

Generalizing Ramanujan's and the Chudnovskys' 1/pi formula (Part 1)

Some years ago, I asked in MSE a question about the Chudnovsky brothers pi formula. Later, I asked in MO a related question. The former was unanswered until a few days ago when L. Miller gave me a ...
Tito Piezas III's user avatar
9 votes
0 answers
218 views

On the first sequence without collinear triple

Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one. ...
Sebastien Palcoux's user avatar
9 votes
0 answers
177 views

Infinite series identities in search of a proof

This comes in relation to the Fishburn numbers. I stumbled on the following relation for which I ask a proof if true. Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then $$\sum_{n=0}^{\infty}\frac{(n+1)zt}{...
T. Amdeberhan's user avatar
9 votes
0 answers
497 views

Can an infinite sum depending on the logarithms of all positive integers be rational or algebraic?

Consider $$ C = \sum_{n=1}^\infty \frac{(-1)^{f_1(n)} f_2(n) \log{n} + f_3(n)}{f_4(n)}$$ The sum converges. $f_1$ is either $0$ or $n-1$, $f_2,f_3,f_4$ are polynomials with integer coefficients and $ ...
joro's user avatar
  • 24.2k
8 votes
0 answers
316 views

A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$

Recently, I found a (conjectural) new series for $\sqrt3\pi$: $$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$ The series converges fast ...
Zhi-Wei Sun's user avatar
  • 14.4k
8 votes
1 answer
359 views

You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...
Dan's user avatar
  • 2,341
8 votes
0 answers
245 views

Switching the order of a summation and replacing a series by its analytical continuation

Background A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
Caleb Briggs's user avatar
  • 1,662
8 votes
0 answers
513 views

Concave and other bounded functions: Series representation and converging polynomials

Main Question Suppose $f:[0,1]\to[0,1]$ is continuous, polynomially bounded, and belongs to a large class of functions (for example, the $k$-th derivative, $k\ge 0$, is continuous, Lipschitz ...
Peter O.'s user avatar
  • 637
8 votes
0 answers
236 views

Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial

This is mostly a reference request. I'm working with complex coefficients, although all I have in mind have integer coefficients. Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
YCor's user avatar
  • 60.1k
8 votes
0 answers
274 views

On rational Ramanujan-type series for $1/\pi$

A Ramanujan-type series for $1/\pi$ is a series of the following form $$ \sum_{n=0}^{\infty} \frac{(\frac12)_n(\frac{1}{s})_n(1-\frac{1}{s})_n}{(1)_n^3} (bn+a) z^n=\frac{1}{\pi}, $$ where $(c)_n=c(c+1)...
Jesús Guillera's user avatar
8 votes
0 answers
191 views

The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1

We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1? I ...
ililiil's user avatar
  • 661
8 votes
0 answers
140 views

Minimum length of sequence such that every integer from 1 to n can be achieved as the sum of some contiguous subsequence

This question literally came to me in a fever dream last night, and it's frustrating me to no end. I'll try to explain it as best I can, but there may not be a satisfying answer; the best outcome ...
Joachim Worthington's user avatar
8 votes
0 answers
557 views

Evaluating $\sum_{k=1}^{\infty} \binom{2z}{z-kn}$

Can we turn this sum into a simpler form ? $$\sum_{k=1}^{\infty} \binom{2z}{z-kn}=\Gamma(2z+1)\sum_{k=1}^{\infty} \frac{1}{\Gamma(z-kn+1)\Gamma(z+kn+1)} \qquad;\qquad (z;n)\in \mathbb C \times \mathbb ...
L.L's user avatar
  • 399
8 votes
0 answers
703 views

Is this 2x2 determinant sequence positive and increasing?

Let $X_1,X_2,X_3$ be three discrete (integer and non-negative valued) random variables with local probabilities $a_k:=\mathbb{P}(X_1=k)$, $b_k:=\mathbb{P}(X_2=k)$, $c_k:=\mathbb{P}(X_3=k)$ and $s_k:=\...
Fancier of Mathematica's user avatar
8 votes
0 answers
986 views

On the sum of consecutive primes and product of first and last

Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$ . $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$ Meaning the sum of set of ...
Shivam Patel's user avatar
7 votes
0 answers
174 views

Permutations which change the value of a convergent series

I'm interested in the following combinatorial problem: What is a necessary and sufficent condition on a permutation $\sigma : \mathbb{N} \rightarrow \mathbb{N}$, so that there exist a summable ...
Et-'s user avatar
  • 71
7 votes
0 answers
419 views

On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$. I. Recurrences involving $\zeta(5)$ In Cohen's 2022 paper, ...
Tito Piezas III's user avatar
7 votes
0 answers
297 views

Gottfried Helms' tetra-eta series

Here Gottfried Helms introduces the following fascinating divergent series $$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$ The terms don't go to zero, so technically the series does not converge ...
Caleb Briggs's user avatar
  • 1,662
7 votes
0 answers
427 views

Dynamics of a curious bijection of $\mathbb N$

The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows: Given an ...
Roland Bacher's user avatar
7 votes
0 answers
133 views

Characterization of tempered distributions from tempered sequences

Let $\mathcal{D}(\mathbb{R})$ be the space of compactly supported and infinitely smooth functions for its usual topology. Let $\mathcal{D}'(\mathbb{R})$ be the topological dual of $\mathcal{D}(\...
Goulifet's user avatar
  • 2,174
7 votes
0 answers
199 views

Fraction of elements in $\mathbb{Z}_n$ satisfying a certain equation

From a question arising in Game Theory, I want to calculate the sequence $$ a_n = \max_{f_A, f_B : \mathbb{Z}_n \to \mathbb{Z}_n} \frac{\# \left\{ (x,y) | f_A(x) - f_B(y) = xy \mod n \right\}}{n^2} $$...
Michael Mc Gettrick's user avatar
7 votes
0 answers
225 views

Is there a connection between the sequence of a finite number of Stieltjes constants and the integer partitions number?

Lehmer 1988 and Keiper 1992 made major progress on evaluating the series: $$\sigma_r = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right) \quad r \in \mathbb{N}$$ where $\...
Agno's user avatar
  • 4,179
7 votes
0 answers
330 views

Do generalizations of the identity $\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) = 1 $ exist?

On p. 263 of Borwein's paper entitled “Computational Strategies for the Riemann zeta function”, the following identity is stated: $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1 . \qquad \...
Max Muller's user avatar
  • 4,485
7 votes
0 answers
197 views

My research paper involves computing additional terms of an existing OEIS sequence. Should I first amend the sequence or publish the results?

In the course of my research I computed terms of an existing OEIS sequence that are currently unknown. Having prepared my paper for publication, I am now faced with a (small) dilemma: Do I first ...
Klangen's user avatar
  • 1,943
7 votes
0 answers
210 views

How to prove the identity $\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{\pi^4}{360}$?

For each $n=0,1,2,\ldots$, the harmonic number $H_n$ is given by $$H_n:=\sum_{0<k\le n}\frac1k.$$ In 2016 I conjectured that $$\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{...
Zhi-Wei Sun's user avatar
  • 14.4k
7 votes
0 answers
308 views

Alternative approaches to Zudilin's proof of Apéry's theorem

The article Wadim Zudilin, Apéry's theorem. Thirty years after, Int. J. Math. Comput. Sci. 4 (2009), no. 1 pp 9–19. (arXiv:math/0202159, with the title An elementary proof of Apéry's theorem) ...
mamiladi's user avatar
  • 417
7 votes
0 answers
205 views

Has this self-similar sequence the ratio $(\sqrt2+1)^2$?

This is inspired by a math.SE question, where an infinite sequence of pairwise distinct natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows: $a_n$ is the smallest number such that $s_n:=...
Wolfgang's user avatar
  • 13.2k
7 votes
0 answers
934 views

A problem of Erdős on convergence of $\sum (-1)^nn/p_n$ and equidistribution of $\pi(n)$ modulo 2

Erdős asked1 whether the series $$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges. Here, $p_n$ denotes the n-th prime. I can show that this series converges simultaneously with the series $\sum_{...
Mustafa Said's user avatar
  • 3,679
7 votes
0 answers
247 views

Why does a polynomial with discriminant $d=-163$ appear in this sequence?

Fact 1: Given the eta quotient, $$x_d= e^{2\pi\rm{i}/48}\,\frac{\eta(\tau)}{\eta(2\tau)}$$ where $\tau=\frac{1+\sqrt{-d}}2$, then $x_d$ for $d=11,19,43,67,163$ are the roots of the simple cubics, $$...
Tito Piezas III's user avatar
7 votes
0 answers
122 views

in search of intepretations and connections for $k$-central binomials

Fix a positive integer $k$. Then, the sequences $$c(n,k)=\frac{k^n}{n!}\prod_{m=1}^{n-1}(1+km)=[x^n]\left(\frac1{1-k^2x}\right)^{1/k}$$ are referred to as "$k$-central binomial coefficients",...
T. Amdeberhan's user avatar
7 votes
0 answers
288 views

An integral for the tribonacci constant and the general case

When I asked for integrals involving the tribonacci constant $T$, user @nospoon gave the nice answer, $$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}} $$ However, the ...
Tito Piezas III's user avatar
7 votes
0 answers
454 views

Analytic expression for the Tsirelson bound of the I3322 inequality?

Finding Tsirelson bounds for Bell inequalities is a well-loved problem in quantum information theory. A famous case where it is still open is for the I3322 inequality. In this paper Pál and Vértesi ...
Mateus Araújo's user avatar
7 votes
0 answers
186 views

How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question. Take the following, nicely symmetrical, telescoping series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum _{n=1}^{\...
Agno's user avatar
  • 4,179
7 votes
0 answers
325 views

About the first decimal of $\sqrt {n!}$

Do we have : $$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$ Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
Med's user avatar
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