Fifty Years of Integer Sequences
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social
Front cover of the first edition of Neil Sloane’s Handbook of Integer Sequences, 1973.
It would be remiss of me to let 2023 pass by without acknowledging that it’s the 50th Anniversary of the appearance of Neil Sloane’s A Handbook of Integer Sequences back in 1973. That was just two years after I completed my PhD, so his reference source has been part of the mathematical landscape throughout my career.
That first edition had just 2,372 entries. In 1995, with Simon Plouffe as a co-author, it rebranded as The Encyclopedia of Integer Sequences, with 5,487 entries. A year later, the list (already doubled in size) moved from print onto the newly established World Wide Web as the Online Encyclopedia of Integer Sequences (OEIS, https://oeis.org). It currently has over a third of a million sequences.
Once it was online, in searchable form, it became a valuable research tool. If a mathematician came across an integer sequence during the course of their research, they could quickly enter the first few terms in OEIS search box, and see what, if anything, was known about that sequence. If the database did not have a match, they could then submit the sequence for future inclusion.
The OEIS can, and has, lead to interesting new connections across disciplines. For example, a few years ago, Lara Pudwell, a mathematician at Valparaiso University in Indiana (also a member of the OEIS Foundation’s board of trustees), who develops algorithms to solve counting problems, entered the following sequence she encountered while studying numerical patterns:
2, 4, 12, 20, 38, 56, 88 ...
The OEIS found just one match, which came from chemistry: the periodic table and the atomic numbers of the alkaline earth metals. Dr. Pudwell consulted with chemists and found there were interesting chemical structures to work with to explain the connection.
Neil Sloane in a fairly recent photograph.
This particular example is recounted by math write Siobhan Roberts in an excellent article about the OEIS published in the New York Times in May of this year.
But before you look there, see if you can find the rule that generates the following sequence
2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, 2006, 2030 …
If you haven’t come across it before, you are likely to find it quite a challenge. It’s called the eban sequence.
Of course, you could quickly find the answer by entering the first ten terms into the OEIS (or by looking up the name on Google). But it’s more fun to reflect on the clue that the secret lies in the sequence’s name. It was invented by Sloane himself around 1990.
Though the OEIS can have valuable use in mathematical and scientific research, it is also a great source of fun for those who like playing with numbers.
Sloane himself is such an individual, as shown by his videos on the mathematics educational video resource Numberphile; such as this one. (The eban numbers are one of the examples he gives in this video.)
For further information about Neil Sloane and his work on integer sequences, see the retrospective article he published in January of this year.
With the three excellent sources I’ve cited available, I’ll stop here and leave you to explore for yourself.
ADDENDUM: Talking of anniversaries and long (really long!) sequences of numbers, mathematician K. P. Hart of Delft University of Technology in the Netherlands just alerted me to the fact that it was exactly 150 years ago that Georg Cantor wrote a letter to Richard Dedekind asking whether the set of real numbers could be put into one-one correspondence with the natural numbers — a question that would soon lead Cantor to create the modern field of Set Theory. See Hart’s recent blogpost on this momentous event.