Questions tagged [oeis]

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

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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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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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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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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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Why this rational approximation $\pi\sim\frac{80249}{25544}$ is not mentioned in OEIS?

I have checked sequence of Denominator of best approximation to $\pi$ with denominator $\le10^n$ in OEIS but I didn't find this rational $\frac{80249}{25544}$ however it is better than $\frac{22}7$, ...
2
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1answer
127 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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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 ...
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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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Is OEIS A248049 an integer sequence?

The OEIS sequence A248049 defined by $$ a_n \!=\! (a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})/a_{n-4} \;\text{ with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$ is apparently an integer sequence but I ...
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1answer
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Counting polysticks on the $n$-cube.

Over at Code Golf Stack Exchange, I put up a challenge asking people to count, among other things, the number of ways to take an $n$-cube and color $k$ (connected) edges up to isometries of the $n$-...
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Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)

(Cross-posted from MathOverflow. See comments there for more background.) The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics ...
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Examples of fusenes that are not polyhexes

OEIS sequence A108070 describes Number of fusenes with n hexagons. OEIS sequence A000228 describes Number of hexagonal polyominoes with n cells. These sequences first disagree at $A108070(7) =...
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80 views

About Harmonic Numbers and a sequence in OEIS

I'd been looking for a series in OEIS and the one that fits better (resembles, at least) is one in this link in Examples: http://oeis.org/A082687. There, as we can see, it is equal to $$H'(2n) = H(...
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1answer
32 views

What is meant with “whose xor-sum is n”?

A sequence look-up led me to sequence A088512 in OEIS.org with description "Number of partitions of n into two parts whose xor-sum is n." I know the "Number of partitions of n into two parts" which ...
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1answer
75 views

About $A_{n+1}=A_{n}+A_{n}^{2}$ and OEIS $A122888$

OEIS A122888 is: $1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 9, 10, 8, 4, 1, 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1, 1, 5, 20, 70, 220, 630, 1656,\dots$ I'm having trouble ...
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47 views

Counting deltahedra with $2n$ faces

A deltahedron is, according to Wikipedia, a polyhedron whose faces are all equilateral triangles. There is only one deltahedron with four faces: the tetrahedron. Likewise, there is only one ...
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1answer
47 views

Are the “distended” numbers precisely the numbers for which no two subsets of their divisors have the same sum?

The OEIS sequence A051772 defines the "distended" numbers as those positive integers $n$ for which each divisor of $n$ is greater than the sum of all smaller divisors. Now, here's a question about ...
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How many of different connected graphs with n vertices? Excluding of isomorphic graphs.

Given $n$ vertices, the question is to answer how many different connected graphs exist? For example, $f(1)=1,f(2)=1,f(3)=2,f(4)=6 \cdots$. Here I exclude the isomorphic graphs. I know there is a ...
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1answer
72 views

Number of strings of $\{0,1,2\}$ : the longest substrings of $1$ is odd-length.

Consider $A^* = \{0,1,2\}^*$. We want to know how many string of length $n$ in this alphabet following such property : all longest substrings of 1-s has odd length(let it be $a_n$). I know that it ...
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83 views

Finding the next term in a sequence: $1,4,17,19,148,.?.$

Define $a(n)$ to be true if $n\mid(1^1+2^2+3^3+...+n^n)$ So $\{n\in\mathbb N\mid a(n)\}=\{1,4,17,19,148,...\}$ What is the sixth term? I checked $1\le n\le 1000$, but did not find a sixth term. ...
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(Why) are $0, 1, 10, 120, 1540, 7140$ the only triangular tetrahedral numbers?

I recently came across entry A027568 of the OEIS, which reports the numbers $$0, 1, 10, 120, 1540, 7140$$ as the set of integers which are both triangular and tetrahedral; that is, each can be written ...
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Are all positive odd integer matchs this three conditions will be a prime number?

The period length of the decimal expansion of $1/n = 2^x$ and $n-1$ divides by the length. The sum of $n = 2^x$. The cycle length of $n = 2^x$. ($x$ is some positive integer) (using $n=23$ as an ...
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The pattern of $n$-th prime minus its reversal in even and odd bases

I recently saw the sequence A265326 on OEIS and also in Brady's Numberphile video Amazing Graphs ft. Neil Sloane The sequence is such: Start from $2$, given the $x$-th prime number and convert it ...
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Computing Remy Sigrist sequence A279125

I am curious to understand how you can compute sequences like this one by Rémy Sigrist https://oeis.org/A279125 You can also see it in action in this numberphile video: https://youtu.be/j0o-pMIR8uk?...
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Where to publish proofs for, for example, the OEIS

TL; DR; Where can I publish proofs that are not groundbreaking nor profound? I recently found an integer sequence in the OEIS. Okay, the OEIS only contains integer sequences, so nothing new so far. ...
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1answer
74 views

What properties does the ratio of two triangular numbers $M$ and $N$ have, assuming $N \mid M$?

This is a follow-up question to this MSE post. Call a number $X$ triangular if it could be written in the form $x(x+1)/2$ where $x$ is a positive integer. Here is my question in this post: What ...
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Is there a specific terminology for numbers which are nontrivial multiples of triangular numbers?

(Note: Please see this new question for the motivation.) A number $T$ is said to be triangular if it could be written in the form $$T=\frac{n(n+1)}{2},$$ where $n$ is a positive integer. Here is my ...
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Binary weight of OEIS sequence A308092.

Preliminaries OEIS sequence A308092 is defined as: The sum of the first $n$ terms of the sequence is the concatenation of the first $n$ bits of the sequence read as binary, with $a(1) = 1$. And ...
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173 views

Number of $n$-bead binary necklaces [OEIS-A000013]

I tried to obtain the number of $n$-bead binary necklaces from my program written in C++. Then, one formula came up when I looked up the number to see if my thought is correct. Number of $n$-bead ...
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1answer
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Intuitive geometric sequences (and where to find them in OEIS)

Wondering where intuitive "geometric sequences" are in OEIS. By intuitive, I mean things that you can easily describe to non-math people, as opposed to more obscure concepts that require some deeper ...
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1answer
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Recurrence for A000670

A000670 contain a formula by Martin Kochanski: Recurrence: $2a(n)=(a+1)^n$ where superscripts are converted to subscripts after binomial expansion - reminiscent of Bernoulli numbers $B_n=(B+1)^n$. ...
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Crossing Number Graphs CN 9-13

At Crossing Number Graphs are minimal cubic graphs with crossing numbers 0 to 8. At Smallest Cubic Crossing Number Graph are some corrections -- 8D and 8E can be drawn with less than 8 crossings, and ...
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1answer
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Number of unlabeled hypergraphs (A003180)

I'm looking for the number of unlabeled hypergraphs on n nodes and stumbled upon the comments of A003180 in OEIS. Can somebody please explain to me how that sequence relates to the number of unlabeled ...
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Why do these two sequences end in an increasing number of zeroes?

The trimorphic numbers are integers whose cubes end in the digits of the integers themselves, such as ${\sf{49}}^3=1176\sf{49}$, and I have discovered something interesting about such integers that ...
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Growth rate of primes vs. prime indexed primes

Looking at the graph of the prime indexed primes on oeis and the graph of the primes, it seems that the first graph has small jumps exactly at the same positions as the second. It's no surprise ...
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1answer
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On an interesting assertion in the OeisWiki page on multiply-perfect numbers

The following (interesting) assertion appears in the OeisWiki page on multiply-perfect numbers: ...
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Gap in spiral sequence

OEIS sequence A272573 describes a sequence generated in the following way: Start a spiral of numbers on a hexagonal tiling, with the initial hexagon as a(1) = 1. a(n) is the smallest positive ...
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For $n,k \in {\mathbb{Z}}^{+}$ (excluding $n=1$), does $\frac{(n+k)!}{n!}$ ever equal $n!$

While investigating an integer sequence, I came across the following two OEIS entries: A094331: Least k such that n! < (n+1)(n+2)(n+3)...(n+k). A075357: a(n) = smallest k such that (n+1)(n+2)...(n+...
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Fewest number of distinct distances between $n$ points in $\mathbb Z^2$

I've been thinking about proving some bounds for the OEIS sequence A319476: $a(n)$ is the minimum number of distinct distances between $n$ non-attacking rooks on an $n \times n$ chessboard. I'd ...
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1answer
169 views

Lexicographically smallest sequence of integers not in the OEIS

A sequence $a_i$ ($i=1,\ldots$) is lexicographically smaller than sequence $b_i$ if either $a_1 < b_1$, or $a_j = b_j$ for $j=1,\ldots, k$ and $a_{k+1} < b_{k+1}$. If I asked for the ...
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1answer
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Expected value of smallest element of descent set

I'm drafting an OEIS sequence, and I've formulated a few conjectures. I was hoping someone here could help me to prove them. Definition Let the descent set of a permutation $\omega \in S_n$ be the ...
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1answer
758 views

OEIS database download [closed]

I am interested in downloading the first 25 integers of each sequence in the OEIS database. My reason for wanting to do so is to find the arithmetic mean of the first integers, the arithmetic mean of ...
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Why would you take the logarithmic derivative of a generating function?

Today, my climbing expedition scaled Mt. Sloane to request the Oracle's Extensive Insight into Sequences. The monks there had never heard of our plight, so they inscribed our query in mystical runes ...
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681 views

Smallest region that can contain all free $n$-ominoes.

A nine-cell region is the smallest subset of the plane that can contain all twelve free pentominoes, as illustrated below. (A free polyomino is one that can be rotated and flipped.) A twelve-cell ...
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The numbers $n$ such that every sum of consecutive positive numbers ending in $n$ is not prime or A138666?

Question: Let $A(n)$ be a finite square $n \times n$ matrix with entries $a_{ij}=1$ if $i+j$ is a prime number; otherwise equals to $0$. I write $|A_n|$ to count the number of $1$'s in $A(n)$ and ...
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Reducing a system of Linear Recurrences

The number of ways to lay unit-length matchsticks on a $2 \times n$ grid (with respect to vertices) appears to be given by OEIS sequence A093129, but I found it a different way, and I'm looking for ...
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Explanation for OEIS $\text{A}000157$ – “Number of Boolean functions of $n$ variables”

I'm having trouble understanding OEIS $\text{A}000157$, not necessarily its description, but rather its terms. I've already visited how many semantically different boolean functions are there for n ...
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1answer
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On a certain OEIS convention

Do you know what it is that those numbers which appear in the upper-right angle of the blue-box mean? I presume they have to do with a certain ranking of the sequences within the site, but I am not ...
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Two miscellaneous questions about equations involving the Euler's totient function and twin primes

This morning I was thinking in equations involving the Euler's totient function $\varphi(n)$ and the sequence of twin primes (see if you want this MathWorld). I am saying similar questions than ...
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Group notation in OEIS

I am wondering how I should interpret the notation that the Online Encyclopedia of Integer Sequences (OEIS) uses for finite groups. An example is this sequence: A008795 In this specific case, what ...