What is it about sequences and mathematicians, really? You’d be hard-pressed to find a number theorist who isn’t fascinated by sequences of numbers, whether they find one in their research or simply conjure one out of thin air. In fact, we’ve all played this game, growing up: throw out a few numbers and ask, what comes next?
For example:
* 1, 3, 5, 7 ... (next presumably 9—the odd numbers).
* 1, 4, 9, 16 ... (25—the squares).
* 0, 2, 6, 12, 20 ... (30—the differences are the even numbers).
* 0, 1, 1, 2, 3, 5, 8 ... (13—the Hemachandra or Fibonacci numbers).
Maybe you think those are pretty straightforward? Maybe yes. But there are endless ways to find patterns and links among the integers, and thus there are endless sequences. In my last column, for example, I mentioned Somos sequences. In the end, it comes down to this: if you can set out a series of numbers, demonstrate a link between them and show that the link is consistent—well, if it’s a sequence nobody has seen before, you might just attract the attention of a lot of mathematicians.
In particular, you might attract the attention of a mathematician called Neil Sloane, chairman of the OEIS Foundation. He worked for years at AT&T and teaches mathematics at Rutgers University, but he’s undoubtedly best known for that OEIS Foundation. That’s the “On-Line Encyclopedia of Integer Sequences". Yes indeed: there actually exists an encyclopaedia of mathematical sequences.
Sloane began compiling sequences nearly 60 years ago, when he was a graduate student at Cornell University. During his research then, he had run across the sequence 1, 8, 78, 944 ... and was searching for a formula that would predict any term in it. Never fear, I will not ask you what number comes next, nor what these numbers represent. But I will tell you: 13,800 is the fifth term, and these are the “normalized total height of all nodes in all rooted trees with n labelled nodes". I have only a fuzzy idea what that means, but I can tell you too that the 20th term is 22,510,748,754,252,398,927,872,000—22 septillion and change.
Sloane couldn’t find any reference to this particular sequence in his books or in the university library. This prompted him to start recording sequences himself, on index cards. That eventually became a book, then another book. In 1996, Sloane moved his collection, by then 10,000 sequences strong, to the internet. It has grown steadily since, adding an average of nearly 40 sequences every day, between 10,000 and 18,000 every year. Today the OEIS lists nearly 370,000 sequences.
But if all that’s impressive, what’s truly remarkable about the OEIS is that it is a gold mine of fascinating information. Here’s a flavour of that.
To start with, if you have a certain sequence of numbers in mind, you can type them in and see if they match something that’s already in the database. I tried “7, 23, 59". Now those three numbers are in the Somos-4 sequence I mentioned in my last column, so I expected that match. But I was simply thrilled to find that they are also part of, wait for it, the Babylonian expansion of π (pi)! That’s what π would look like if we used, as ancient Babylonians did, 60 (sexagesimal) and not 10 (decimal) as a base for our number system.
What’s more, those three are also part of 11 other sequences —besides a few related to Somos-4—in the OEIS database. It happens that most of those 11 involve prime numbers—like OEIS Sequence #A084710, “the smallest prime greater than the previous term such that the difference of successive terms is a distinct square", in which the third, fourth and fifth terms are 7, 23, and 59.
Then there are sequences that are interesting for different reasons. The primes are there, of course. That may not excite you. But the Mersenne primes are also there—numbers formed by raising 2 to the power of certain primes (themselves forming a separate sequence) and subtracting 1. That may not excite you either. But here’s what might: the largest prime number we know is a Mersenne—raise 2 to 82,589,933 and subtract 1, producing a number with 24,862,048 digits. Not just that, the next five largest prime numbers we know are Mersennes, too. Not just that, the last seventeen times this largest-prime record has been broken, it’s been by Mersennes. In fact, you’re welcome to join the Great Internet Mersenne Prime Search, a collaborative effort to find ever larger Mersennes. For the time being, though, OEIS sequence A000668 will show you the first 12 Mersennes.
Sequence A250000 is about the charmingly-named Peacable Queens. It answers this question: On a square chessboard of a given size, how many white and black queens can you place so that they don’t directly attack each other? On a 3x3 board, you will only be able to place 1 queen of each colour. On a regular 8x8 board, 9 queens each will ensure the peace, and so on.
Or you can pick a sequence and challenge yourself: What’s the rule that generates this one? Naturally, purely mathematical rules will probably be difficult. But there are others that have appeared here and there, as puzzles for non-mathematicians like me. My favourite of these is sequence A006751, whose first few terms are: 2; 12; 1112; 3112; 132112; 1113122112; 311311222112; and 13211321322112. Work the rule out for yourself and then write down the next term. Here’s a hint: Do not insert the usual commas in those numbers.
Finally, there are pairs of sequences defined totally differently that seem to be identical, but that are actually not. Take A010918, the “Shallit sequence", and A019484—never mind their esoteric definitions, but I assure you they are different. Both start thus: 8; 55; 379; 2612; 18,002; 124,071; 855,106; 5,893,451; 40,618,081; 279,942,687 ... and in fact OEIS tells us that their first 11,055 terms match. The 11,056th terms—each 9,270 digits long—differ by 1, and then they diverge.
Right there, you have a question that mathematicians will yearn to answer: Why do these different definitions produce 11,055 identical terms, and then differ? Right there is one reason the OEIS is such a mathematical treasure trove.
Makes me wonder: Is there a sequence made up of the counts of such identical terms in pairs of series, 11,055 being an entry? Would it make an interesting submission to the OEIS?
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun.
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