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Math Trek

The pattern collector

The Encyclopedia of Integer Sequences outgrows its creator

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When Neil Sloane was a young man, he started collecting objects he found beautiful. A common enough preoccupation perhaps, except for the particular objects Sloane chose: number sequences.

He has classical sequences that have captivated mathematicians for millennia, like 2, 3, 5, 7, 11, 13…, the prime numbers. He has tricky sequences like 1, 1, 2, 5, 14, 38, 120, 353…, the numbers of different ways of folding ever-longer strips of postage stamps. He has dull yet fundamental sequences like 0, 0, 0, 0…, the zero sequence.

He even has sequences that might wreck your life. Read this one at your peril: 0, 1, 3, 6, 2, 7, 13, 20, 12,… (A005132).

In fact, he now has nearly 200,000 number sequences in a searchable online database, and his personal obsession has become a treasure for the entire mathematical community. Mathematicians, computer scientists, physicists and other researchers search (by number or sequence name) his On-line Encyclopedia of Integer Sequences (www.oeis.org) thousands of times every day. When they don’t find the sequence they’re looking for, they email Sloane to suggest an addition. He receives and personally reviews an average of 45 such emails per day — an influx that has steadily grown over the last four decades. Recognizing that he can no longer keep up with the flood, he is now turning the site into a wiki, with more than 70 associate editors taking over his duties.

“The OEIS has really changed the way mathematicians and other scientists work,” says Doron Zeilberger of Rutgers University. “It’s useful in many ways, but one of the most interesting is that it often reveals surprising connections. Usually there’s a deep reason, and then the challenge is to understand why.”

The OEIS — or simply “Sloane,” as it is more frequently called — does far more than merely identify sequences. It is much like the Oxford English Dictionary: The OED provides the earliest quotations for each usage of a word, and the OEIS similarly provides a sequence’s full “life story.” Along with listing the numbers that form the beginning of a sequence (sometimes hundreds of thousands of them), it gives all the different known ways to generate the sequence, lists references to the sequence in the scientific literature, links to any sites with information about it, cross-references related sequences, provides a graph of the sequence, and even offers a way to listen to the sequence.

By some measures, the OEIS is an even larger undertaking than the OED. The OED has about 220,000 entries — a bit larger at present than the OEIS. But it appears that there may be no end to human ingenuity to come up with number sequences. The OEIS website says, “It is hoped that eventually the database will include every (interesting) number sequence that has ever been published.” But Sloane’s optimism has faded under the onslaught of email: “I’m afraid,” he says, “that the rate of growth will continue to increase.”

Sloane, now a mathematician at AT&T, got lured into his quixotic quest by an elusive sequence. As a graduate student, he studied neural networks (then called “perceptrons”), a brand-new idea at the time. A neural network is an algorithm that functions like a human brain, with artificial “neurons” connected by “synapses” that learn to do computations by adjusting their connections. Because so little was then understood about neural networks, Sloane figured he’d ask the most basic possible questions initially and began by considering only neural networks having connections that formed no loops, giving them a tree-like structure. Then he asked: If he picked a random node on a random tree containing n total nodes, how far on average would the node be from the root of the tree? He figured it out for the first few n and got this sequence: 0, 1, 8, 78, 944, 13800, 237432, . . .

“That sequence is still engraved in my memory,” Sloane says, burned into place by frustration. He wanted to know how quickly the sequence grew as n got bigger, but just looking at the sequence, he couldn’t guess. Nor could he figure out a formula to generate the sequence. Not only that — the sequence didn’t seem to appear in any combinatorics books. As he thumbed through the books past sequence after sequence, he was seized by the conviction that some day, he’d need one of these other sequences and wouldn’t be able to find it. So he decided to start keeping a list, putting each sequence on a card. Nine years later, in 1972, he turned his stack of nearly 2,400 sequences into a book.

Mathematicians were overjoyed. “There’s the Old Testament, the New Testament and the Handbook of Integer Sequences,” one commenter wrote.

The book led to a sequel, the sequel led to the website, and now the website is leading to the wiki. Sloane never did, however, find the sequence he was initially looking for in any books. Eventually, he and the late John Riordan of Bell Labs managed to derive the formula, and it became sequence A435.

Sloane continues to delight in the sequences that come his way, sometimes spending months researching their properties. For example, years ago Colombian mathematician Bernardo Recamán Santos sent Sloane the sequence 0, 1, 3, 6, 2, 7, 13, 20, 12,… (A005132), which became Sloane’s all-time favorite sequence. Unlike many sequences in the database, the Recamán sequence seems to be just a curiosity, unconnected to other mathematical questions. But Sloane loves it anyway. “Contemplation of such wonderful discoveries,” he says, “provides a welcome escape from the troubles of our planet.”

Though a preliminary version of the wiki is online, the transition has been slowed by the inability of current wiki software to search for sequences of numbers. Sloane has already set up a non-profit foundation to manage the database and transferred the intellectual property rights to it. That moment was wrenching for Sloane: “It felt a bit like giving away one’s children.”

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