Questions tagged [oeis]

For questions related to the On-Line Encyclopedia of Integer Sequences.

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Counting endofunctions by inclusion–exclusion

While leafing through the OEIS, I noticed the following conjecture (from Werner Schulte, OEIS A000312 ): For integer $n \ge 0$ $$ \sum_{k=0}^n (-1)^{n - k} { n \brack k }{n+k \brace k} = n^n . $$ ...
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0 votes
0 answers
57 views

Is there really no way to generate infinitely many primes?

Is there really no way to generate infinitely many primes? A previous answer for someone asking about the Infinite generation of primes, says: There is no exact way to generate primes continuously. ...
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  • 205
9 votes
1 answer
102 views

Counting permutations $\pi \in S_n$ such that $\pi(i) \neq i$ and $\pi(i) - k \not\equiv i \bmod n$.

Given integers $n > 2$ and $1 \leq k < n$, I'm interested in the set of permutations $$ \{\pi \in S_n : \pi(i) \neq i \text{ and } \pi(i) - k \not\equiv i \bmod n\}. $$ When $k = 1$, these are ...
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  • 4,871
7 votes
2 answers
158 views

Asymptotic for $\sum_{k=1}^n k^n$

Consider the OEIS sequence A031971, which is defined as: $$a_n=\sum\limits_{k=1}^n k^n\quad\color{gray}{(1,\,5,\,36,\,354,\,4425,\,67171,\,1200304,\,.\!.\!.\!)}\tag{1}$$ I'm interested in the ...
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7 votes
0 answers
43 views

How many lattices does it take to cover a regular $n$-gon?

Given some positive integer $n\ge 3$, we can ask how many 2-dimensional lattices $L_1,\ldots,L_k$ are required such that their disjoint union contains all vertices of a regular $n$-gon. (We don't ...
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1 vote
1 answer
44 views

Question about the least prime $p$ such that $p+2n$ is also prime.

In the sequence Least prime $p$ such that $p+2n$ is also prime page (A020483) on OEIS, it says: If $a(n)$ exists, $a(n) < 2n$ What does it mean? At first it sounds like if both $p$ and $2n+p$ are ...
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3 votes
1 answer
61 views

Pattern in number of digits in $(10^n)!$

I was observing the OEIS sequence A061010 and its values up to $n=1000$, and it seems that the pattern is as follows. For $n$, $f(n)$ is for the form string of $(n-1)$+(pattern of repeating digits ...
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0 votes
1 answer
67 views

Recurrence relation to solve

I have one problem that can be transformed into recurrence relation shown below, I guess $a_{n} = O(n)$. But I cannot solve the analytic form of $a_{n}$, can anyone help me on this? $$(n-2)a_{n+1}-(n-...
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  • 103
0 votes
1 answer
53 views

Understanding formula for OEIS A334742: a(A033638(n)) = a(A002620(n))

How do I interpret the formula for OEIS A334742 (https://oeis.org/A334742), which is given as: $$a(A033638(n)) = a(A002620(n)) \,\,\mathrm{for}\,\, n > 1.$$ Since $$A002620(n)= \Bigl\lfloor\frac{n^...
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  • 363
3 votes
2 answers
54 views

An analogue to OEIS for graphs and ordered pairs?

Does anyone know if such a thing exists already? I was able to find a number of seemingly very comprehensive encyclopedia-like reference pages on many species of graphs, but it seemed that the use ...
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  • 4,969
2 votes
0 answers
87 views

Lyndon words over the binary alphabet that begin with "00"

Lydon words over the binary alphabet are strings of $1$s and $0$s such that all rotations of the word are strictly lexicographically later. I've been looking at Lyndon words of length $n$, which are ...
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  • 4,871
2 votes
0 answers
124 views

Understanding first hitting time probabilities for a $2d$ random walk

The probability for a random walk that starts at the origin to return to the origin for the first time after exactly $2n$ steps has been solved on this site a couple times for the $1$d random walk. ...
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1 vote
0 answers
69 views

Probability distribution of number of returns after $2n$ steps of $1d$ random walk

The expected number of returns of a simple random walk in $1d$ after $2n$ steps is a well known problem, with answer $\sum_{m=1}^n \frac{\binom{2m}{m}}{2^{2m}}$. A question that requires more ...
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2 votes
1 answer
70 views

OEIS A333017, or something else entirely?

I am trying to find out how to calculate the number of different relationships there are between all the non-empty cliques that can be made of a given number 'n' of individuals. Doing this manually (...
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  • 173
1 vote
1 answer
81 views

Regarding OEIS and integer sequences

The OEIS, aka Sloane, is an online repository of over $3,20,000$ integer sequences. Now, suppose I have an integer sequence, say, $a_1, a_2, a_3, ...$ I look for this sequence in OEIS. The sequence ...
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1 vote
1 answer
46 views

Does replacing each prime factor of an odd abundant number with the preceding prime always give another abundant number?

Given an odd abundant number $n$, if one replaces each prime factor of $n$ with the preceding prime while maintaining the same multiplicity (which gives A064989($n$)), does one always get another ...
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2 votes
0 answers
45 views

Generation function for the finite Coxeter group of type $D_k$

Basically I'd like to know how to derive the generation function for the finite Coxeter group of type $D_k$ to be familiar with notes in OEIS A162288: According to formula section: 'The growth series ...
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1 vote
1 answer
81 views

Question about A051002 in OEIS

I have a question about sequence https://oeis.org/A051002 in the OEIS database with first elements 1, 1, 244, 1, 3126, 244. I searched however for ...
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5 votes
1 answer
85 views

A recursive approach to Josephus problem (OEIS 225381)

Taking integer $x$ as input: If $x$ is even, then return $x/2$ as series item, giving $x/2$ as input to continue the loop; Else return ceiling($x/2$) as series item, terminate the loop; Finally we sum ...
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  • 61
1 vote
0 answers
42 views

Series $\sum_{k=0}^\infty \frac{k^u}{k!}$---something known? [duplicate]

Define $f(u)=\sum_{k=0}^\infty \frac{k^u}{k!}$ for natural numbers $u$. Using maple I can find the sum for specific values of $u$, but there is no general answer (unsurprisingly). All values I found ...
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1 vote
1 answer
149 views

Questions about the sequence OEIS A059650

Comparing the sequences OEIS A001951 $$a(n) = \lfloor n\sqrt{2}\rfloor = \lfloor \sqrt{2n^2 }\rfloor = \lfloor \sqrt{2*Square }\rfloor:\quad \{0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, .....
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2 votes
0 answers
46 views

Squares in the hypercube and humps in Motzkin paths

The number of squares with vertices coming from the vertices of the $n$-dimensional hypercube $\{0,1\}^n$ is given by $$ 2^{n-3}\sum_{j=1}^{n} \binom{n}{j}\binom{n-j}{j}. $$ The $0$-indexed sequence ...
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  • 4,871
0 votes
1 answer
75 views

Brun's theorem - notation

The series $1,3,6,9,12,14$ represent the first six terms of the series A168045 on OEIS. The formula for this says $a_n = 2n + n / log n + O(n / (log n)^2)$. I do not understand this notation. What ...
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  • 154
8 votes
1 answer
124 views

Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb

Drake Thomas and I have proposed a sequence A343909 to the On-Line Encyclopedia of Integer Sequences (OEIS), which counts "generalized polyforms": generalizations of free polyominoes (Tetris ...
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  • 4,871
6 votes
1 answer
98 views

The Sierpiński triangle and the number of $(0,1)$-polynomials $p(x)$ where $p(x)^2$ has largest coefficient $k$.

My Twitter bot @oeisTriangles randomly selects an OEIS "table"-style sequence and draws an image, where even terms are light-colored and odd terms are dark-colored. Today it tweeted an image ...
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  • 4,871
1 vote
1 answer
46 views

OEIS sequence A308092: run lengths of bits and run lengths of an auxiliary sequence.

In February 2018, when the On-Line Encyclopedia of Integer Sequences (OEIS) was approaching it's 300,000th sequence, Neil Sloane sent an email out to the SeqFan mailing list announcing hand-picked ...
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  • 4,871
5 votes
0 answers
35 views

Chromatic polynomial of the cross-polytope and denominators of convergents to e.

Let $C_n$ denote the $1$-skeleton of the $n$-dimensional cross-polytope, and $\chi_{C_n}(x)$ be the chromatic polynomial of $C_n$. This is equivalent to the way of coloring the $(n-1)$-dimensional ...
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  • 4,871
4 votes
1 answer
65 views

Packing density of 2143 and crossing number of complete graph

I've observed a numerical coincidence, and I'd like to know if there's something deeper here, or if I'm just observing a numerical coincidence due to Richard Guy's Strong Law of Small Numbers. I've ...
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  • 4,871
5 votes
3 answers
161 views

Greatest number of occurrences of the pattern 4213 in a permutation.

Statistic St000750 in the FindStat database is a map $\operatorname{St000750} \colon S_n \rightarrow \mathbb N_{\geq 0}$ given by The number of occurrences of the pattern $4213$ in a permutation. ...
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  • 4,871
-1 votes
1 answer
63 views

What is the sum of the series with terms 1/(2/(3/(4/(5/(6/...?

I wondered how continually compounding a fraction in the denominator would behave as the successive denominators increment. What I was doing was essentially: $\dfrac 1 {\dfrac 2 {\dfrac 3 {\dfrac 4 {\...
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3 votes
1 answer
96 views

Does the number of permutations in $S_n$ with major index equal to $k$, satisfy a degree $k$ polynomial?

I'm interested in fixing $k$ and finding a formula, $M_k(n)$, for the number of permutations $\pi \in S_n$ such that $\operatorname{maj}(\pi) = k$, where $\mathrm{maj} \colon S_n \rightarrow \mathbb ...
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  • 4,871
15 votes
1 answer
292 views

Mistake in OEIS A103904?

The sequence OEIS A103904 is described as Number of perfect matchings of an $n \times (n+1)$ Aztec rectangle with the third vertex in the topmost row removed. Definition of $M \times N $ Aztec ...
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  • 317
26 votes
0 answers
421 views

How many "prime" rectangle tilings are there?

Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, ...
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16 votes
1 answer
317 views

How many combinatorially distinct ways are there to tile an equilateral triangle with $k$ $60^\circ-120^\circ$ trapezoids?

I believe there is exactly one way (up to combinatorial equivalence) to arrange 3 trapezoids with angles of $60^\circ$ and $120^\circ$ into an equilateral triangle: With $4$ trapezoids, I see two ...
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9 votes
4 answers
4k views

A $2021$ problem: $20\sim 21$ and $43\times 47$

Notice that $2021$ is a concatenation of consecutive integers: $20\sim 21$ Also $2021$ is a product of consecutive primes: $43\times 47$. What is the next number with both of these properties? $...
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3 votes
1 answer
157 views

Contradiction between OEIS and factordb.com

OEIS A014545 says 1+13494## is a prime number but factordb.com says it is composite, where n## is the product of first n primes on factordb. Which is correct? edit: Sorry if my question is not ...
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2 votes
1 answer
123 views

Ratio of obtuse triangles to acute triangles in the square.

I came across OEIS sequences A190020 and A190019, and I noticed that they seemed to grow at a similar rate. A190020: Number of obtuse triangles on a (n X n)-grid (or geoboard) A190019: Number of ...
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  • 4,871
1 vote
0 answers
44 views

Big O notation for OEIS A015617

I am having difficulty finding the exact Big O notation for the size of the search space of OEIS A015617 as a function of n. I am researching a theory that A015617 may have useful applications in ...
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  • 247
2 votes
0 answers
81 views

The Length of the Longest Maximal Chain Among The Set of All Integer Partitions with Dominance Ordering

Question: I write the following infinite table: $$T= \text{ }\begin{matrix} 1\\ 1&1\\ 1&2&1\\ 1&1&3&1\\ 1&1&1&4&1\\ 1&2&1&1&5&1\\ 1&1&...
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4 votes
2 answers
121 views

Non-values of $a^3 + b^{c^3} - 3 a b c$ for $a,b,c\in\mathbb{N}$ like $n$ for prime $19n+17$?

I stumbled on a curious case of Richard Guy's Strong Law of Small Numbers because of a typographical error. I intended to type a^3 + b^3 + c^3 - 3 a b c and look at ...
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2 votes
0 answers
59 views

Are (1,2) and (4,5) the only two consecutive pairs in A003592?

All odd numbers in A003592 are powers of $5$, so this is equivalent to finding all $n \ge 0$ such that $5^n = 2^m+1$ or $2^m - 1$ for some $m \ge 0$. By quick brute force computation I cannot seem to ...
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  • 176
3 votes
2 answers
121 views

Number of convex polyhedra whose faces are regular polygons and whose largest face is an $n$-gon

I'm trying to count up the number of convex polyhedra whose faces are regular polygons and whose largest face is an $n$-gon. (I.e. either a uniform polyhedron or a Johnson solid.) If I've done my ...
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  • 4,871
2 votes
0 answers
141 views

Proof of the remarkable formula for the n-th non-square?

The OEIS's A000037 entry makes the remarkable claim that every non-square number is given by the sequence $$a(n) = n + \Big\lfloor \frac 12 + \sqrt n\Big\rfloor$$ After looking through the entry, I ...
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  • 361
0 votes
1 answer
50 views

On Hypergeometric Series and OEIS Sequence

I have been searching an integer sequence in OEIS. The sequence is the following: OEIS A321234 (https://oeis.org/A321234) . So far, so good. However, this sequence is the denominator of a ...
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  • 434
3 votes
0 answers
41 views

Is there any website like OEIS for special polynomials

I would like to know if there is any kind of website (like OEIS) in which we can search for special known polynomials. For example, we put the coefficients of Legendre's and then the website gives us ...
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  • 434
1 vote
2 answers
103 views

Explicit closed formulas for A056542 and A079751?

Consider the recurrence $B_1 = 0$, $B_n = nB_{n-1} + 1$ for $n\ge 1$ as defined by http://oeis.org/A056542 or by R. Sedgewick, Permutation generation methods, Computing Surveys, 9 (1977), 137–164. How ...
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2 votes
1 answer
183 views

Does a graded poset on $\mathbb{N}_{>0}$ generated from subtracting factors define a lattice?

Consider the partial ordering of positive integers with covering relations $n - \frac np \lessdot n$ for all prime divisors $p \mid n$. This defines a graded poset with $A064097$$(n)+ 1$ rank levels ...
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  • 4,871
11 votes
0 answers
1k views

Fractal Pattern from Queen's Move Construction

This question relates to the OEIS sequence A279212. Fill an array by antidiagonals upwards; in the top left cell enter $a(0)=1$; thereafter, in the $n$-th cell, enter the sum of the entries of those ...
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17 votes
2 answers
726 views

The equation $\sigma(n)=\sigma(n+1)$

In OEIS, the solutions of $$\sigma(n)=\sigma(n+1)$$ where $\sigma(n)$ denotes the sum of the divisors of $n$ including $1$ and $n$ , are shown upto $n=10^{13}$ The entry can be found already by ...
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  • 78.7k
61 votes
1 answer
3k views

Is OEIS A248049 an integer sequence?

The OEIS sequence A248049 is defined by $$ a_n \!=\! \frac{(a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})}{a_{n-4}} \;\text{with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$ is apparently an integer ...
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