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What Number Comes Next? The On-Line Encyclopedia of Integer Sequences Knows.

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Some numbers are odd:

Some are even:

After which there are the puzzling “eban” numbers:

What number comes next? And why?

These are questions that Neil Sloane, a mathematician of Highland Park, N.J., likes to ask. Dr. Sloane is the founding father of the On-Line Encyclopedia of Integer Sequences, a database of 362,765 (and counting) number sequences defined by a precise rule or property. Akin to the prime numbers:

Or the Fibonacci numbers — every term (starting with the third term) is the sum of the 2 preceding numbers:

This 12 months the OEIS, which has been praised as “the master index to mathematics” and “a mathematical reminiscent of the FBI’s voluminous fingerprint files,” celebrates its fiftieth anniversary. The unique collection, “A Handbook of Integer Sequences,” appeared in 1973 and contained 2,372 entries. In 1995, it became an “encyclopedia,” with 5,487 sequences and a further writer, Simon Plouffe, a mathematician in Quebec. A 12 months later, the gathering had doubled in size again, so Dr. Sloane put it on the web.

“In a way, every sequence is a puzzle,” Dr. Sloane said in a recent interview. He added that the puzzle aspect is incidental to the database’s essential purpose: to prepare all mathematical knowledge.

Sequences present in the wild — in mathematics, but additionally quantum physics, genetics, communications, astronomy and elsewhere — might be puzzling for varied reasons. Looking up these entities within the OEIS, or adding them to the database, sometimes results in enlightenment and discovery.

“It’s a source of unexpected results,” said Lara Pudwell, a mathematician at Valparaiso University in Indiana and a member of the OEIS Foundation’s board of trustees. Dr. Pudwell writes algorithms to unravel counting problems. Just a few years ago, thus engaged, she entered into the OEIS search box a sequence that arose while studying numerical patterns:

The one result that popped up pertained to chemistry: specifically, to the periodic table and the atomic numbers of the alkaline earth metals. “I discovered this perplexing,” Dr. Pudwell said. She consulted with chemists and shortly “realized there have been interesting chemical structures to work with to clarify the connection.”

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