“This is one of my top favourites of all time,” said Neil. Term 5: Since we have seen a 0 before, two steps back, the fifth term is 2.Term 4: Since we haven’t seen a 1 before, the fourth term is 0.
Term 3: Since we have seen a 0 before, one step back, the third term is 1.Term 2: Since we haven’t seen 0 before, the second term is 0.Term 1: The first term is 0 by definition.But if the number k has appeared previously in the sequence, then you count the number of terms since the last appearance of k, and that number is the following term. The rule here is that you start with 0, and whenever you get to a number you have not seen before, the following term is a 0. I caught up with Neil via Skype and asked him whether he had any new favourite sequences from the last few years. “Neil did not prove Fermat’s Last Theorem, nor did he prove the Poincaré Conjecture, nevertheless, the impact of on today’s (and tomorrow’s!) mathematical research far surpasses that of any living mathematician.” Professor Doron Zeilberger of Rutgers University believes that Sloane’s collection of sequences has made him the world’s most influential mathematician. It is a marvellous resource, which gets about nine million hits a month, and it has turned Neil – who was born in Wales, brought up in Australia and has spent his adult life in the USA – into the sequences superstar. The OEIS still grows by about 40 sequences a day. Today the OEIS approaches an incredible 250,000 sequences!Įvery day mathematicians around the world send Neil new sequences, which are then approved or rejected for inclusion by him and his team of 20 editors-in-chief and 100 associate editors. In 1996 Sloane put his collection, by then 10,000-strong, on the web, renaming it the OEIS. He was only interested in integer sequences, which are those made up of whole numbers and negative numbers and zero.īy 1973 he had 2,400 sequences, which were published as a book: A Handbook of Integer Sequences. In the 1960s he started to write his favourite ones on file cards. Some people collect stamps, some collect fossils but Neil has always collected sequences. The OEIS is one of the best-known mathematical databases on the web and this week it is the subject of a conference at Rutgers University, timed to coincide with its 50th birthday and the 75th birthday of its founder Neil Sloane. (The first entry is about groups and too complicated for discussion here). The Kolakoski sequence is prestigious for another reason: it is the second entry in the Online Encyclopedia of Integer Sequences (OEIS). It is the only sequence that does this except for the same sequence with the initial 1 deleted. In other words, the Kolakoski sequence describes the length of runs in the sequence itself. The Kolakoski sequence: the length of each run recreates the sequence. If we start at the beginning of the sequence, then, as illustrated below, the length of the runs recreates the original sequence. They appear either in a run of one, or in a run of two. Mathematicians drool over this sequence, because of the pleasingly self-referential way it is defined. Prime numbers and Fibonacci numbers are well known throughout general culture.īut have you ever heard of the Kolakoski sequence?ġ, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, … For example, the prime numbers – the numbers that can only be divided by themselves and 1 – form a sequence:Īnd there is the Fibonacci sequence, for which each term is the sum of the previous two terms: It is just an ordered list of numbers whose terms can be described. Today I want to write about favourite sequences.īefore we get to some, here’s what a sequence is. I’ve written before about favourite numbers.
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