OEIS solves real-world problems! One time I needed to optimize accesses to a data structure in my code. By trying out a few examples, I found the optimal order of iterations followed a particular number sequence, but I had no idea on how it could be deduced or calculated analytically. I entered the few examples I had into OEIS and found the solution. The solution involves graduate-level combinatorics and there was no way that I could solve it.
I suspect this happens more often than anyone realizes.
I really want to read or write the sci-fi story where aliens ping our intelligence with some obscure sequence and they think we're geniuses because thanks to the OEIS we're all like "Oh, yeah, that's just A260151, numerators of coefficients c(n) in asymptotic expansion of Sum_{m=1..k} sqrt(m) ~ zeta(-1/2) + (2/3)*k^(3/2) + (-1/2)*k^(1/2) + Sum_{n=0..inf} c(n)/k^(2*n+1/2)., here's the next 50 terms" https://oeis.org/A260151 in about two minutes.
I had a very similar experience—discovering Stirling numbers of the second kind by solving a computer science problem through brute force, then plugging the results into OEIS, then reverse-engineering an elegant solution to the problem. Fantastic resource.
I can't give any detail, but the solution to my problem involves k-variate k-nomial multinomial coefficients [1]. It's a generalization of the binomial coefficients to high-order scenarios such as trinomial coefficients, quadrinomial coefficients, etc. In other words, from a Pascal triangle to a Pascal pyramid, to a Pascal hyperpyramid.
Multinomial coefficients are also used to define the multinomial distribution in probability which models e.g. the problem of "if I throw a fair die n times, how likely is it that I get exactly x1 ones, x2 twos, ..., x6 sixes?" - which is a generalisation of the binomial distribution in which there are only two cases.
This is not "graduate-level", it's a straightforward generalization of something that all school age kids learn. (Which is not too say that it's trivial to obtain an expression. I'm just saying it's not conceptually advanced)
That's like arguing all math is addition (it's just a bunch of sum signs), therefore all math is not actually graduate level. It's a straightforward generalization of something that all school kids learn.
Also, math achievement in America is horrible. 40% of students leave 12th grade at "Less than Basic". Only 2-3% leave 12th grade at "Advanced." Must not be very straightforward. (21% "Proficient", 35% "Basic" in 2019)
Fun fact: The sequences of ceiling( 2 / (2^(1/n)-1) ) and floor( 2n / log(2) ) have identical elements for a very long time. The first time they differ is n = 777451915729368.
Oh man. I love this encylopedia. Once, I found a little website that had a series of increasingly difficult, Project Euler style coding problems. After a while, it turned out to be a google recruiting tool. But while I was playing the games, I had to use the OEIS more than once. It’s such an obscure, nerdy tool. I realky like it.
OEIS is fun but the bare homepage probably isn't too helpful. I highly recommend Sloane's memoir about the history of OEIS: http://neilsloane.com/doc/HIS50.pdf "_A Handbook of Integer Sequences_ 50 Years Later", Sloane 2023
The OEIS is a curated selection of integer sequences. It only includes those sequences deemed interesting (usually mathematically, but also recreationally, and occasionally for some other reason). It's easy to generate a new sequence from an old one, say B(n) = A(n) + 11 for some sequence A. But if you submit that it will be rejected because it isn't interesting. (Trying to come up with an OEIS-worthy sequence is difficult and edifying, and a great rainy-day activity.)
But even if the OEIS had no standards and included every imaginable sequence, it still wouldn't include more than a vanishingly small fraction of the total set of all integer sequences. This is because almost all integer sequences are infinitely long and cannot be specified. There just aren't enough words!
At one point a project in a research group I'm part of did automatic parsing of the entire OEIS to find relations between different existing sequences. Using a very simple approach, they found ~300 000 000 relations (e.g. meaning one sequence can be expressed as some combination of other sequences); see section 4 of [1].
However they submitted only three (!) of those back to OEIS.
Even with 130 reviewers on the OEIS side, submitting all of those relations would have basically been a Denial-of-service attack on the review process.
Yes, one useful application of Cantor's theorem is to show that anything claiming to enumerate all integer sequences must fail to do so. That's assuming that the sequences can be infinite; if it were the Online Encyclopedia of Finite Integer Sequences, then it could succeed at enumerating all of them.
(As for the diagonal argument, make sure that the ith value of the counter-sequence DIFFERS from the ith value of the ith sequence. A sequence whose ith value matches the ith value of the ith sequence doesn't produce a contradiction, and could in fact be part of the encyclopedia.)
You don't even have to repeat the diagonal argument. There's a one-to-one correspondence between subsets of the natural numbers and sequences of elements from {0,1}. The i-th element is 1 if i is in the set, otherwise it's zero.
Russ Cox (who recently did a deep dive into the xz attack shell script[0][1]) of Bell Labs Plan9 and Google golang fame re-wrote the OEIS in golang in the summer of 2009[2][3] and now serves as the foundation's president[4]
> "Here, however, we ran into a very serious problem. In the summer of 2009, when we tried to get the OEIS working as a wiki, we discovered that the Mediawiki software was not capable of handling the kind of queries that arise in looking up sequences. This was a disaster."
> "It took us over a year to resolve this problem. In the end, Russ Cox completely rewrote all the programs needed to maintain the database and answer queries - a huge task! NJAS's colleague David Applegate has also been of enormous help in getting the new system working. As a result of their work, the new OEIS was finally launched on November 11, 2010. It is now possible for anyone in the world to propose a new sequence or an update to an existing sequence. To do this, users must first register. A group of about 130 editors has been formed, whose job it is to review submissions before they become a permanent part of the OEIS."
> "So, after nearly two years of struggle, the OEIS was finally able to operate without NJAS having to approve every change. After 46 years of running the database, this came as a great relief to him."[2]
> "It's true. The original software was an email auto-reply implemented in shell. The first web version of the software was CGI invoking roughly the same shell script. I didn't have anything to do with those. The next web version (which I wrote in ~2006) was CGI invoking C with an mmap'ed index file. The third web version (which I wrote in ~2010 and is the one running today) is a Go HTTP server, fronted by Apache."
> "Note that I'm only talking about the software for the "interactive" UI, not the database itself. The database itself goes back to punched cards and the original interactive UI was a pair of published books (first A Handbook of Integer Sequences, and then the Encyclopedia of Integer Sequences)."[5]
> It is easy to fit formulas to the first six terms, in a million different ways, but none of them are of any interest (the seventh term can be chosen to be any number you wish). - N. J. A. Sloane, Oct 22 2017
A microcosm for the "plot" of that show, where the writers just pulled some new twist out of their backsides every episode!
There seems to be a few python libraries out there that will spit out a subset of these sequences. I know that's pretty much the exact opposite of how you're supposed to use this data, but I still think it's neat.
Usefully, there's a "PROG" entry for many sequences on this site showing one or more ways of generating the sequence.
I love this site. I have used it many times to help me solve- or shine a light on some problems I have worked on. I visit "my" sequences from time to time
Out of all the professionals, Neil and company are the only community willing to publish my discoveries despite the fact I'm amateur. Looking up A297189 and learning it wasn't known to approximate 7^(1/2) was a thrill.
I like the phrasing used by https://houseofgraphs.org/ — while there are an infinite number of mathematical objects in any category, there are only "a few thousand that can be considered really interesting."
Happy to have authored A094777, Number of legal positions in Go played on an n X n grid (each group must have at least one liberty), [1] and A269417, Number of Go games on n X n board with no repeating position and suicide allowed. [2].
Thanks Neil Sloane and OEIS.