Famous Number Sequences and Their Histories

What counts as a sequence

A number sequence is an ordered list of numbers that follows a rule: the even numbers, the squares, the primes. Some are given by a formula, some by a recurrence in which each term depends on earlier ones, and some by a property that a number either has or lacks. Much of the interest in the sequences below comes from the gap between a simple rule and surprisingly deep behaviour. This overview sets out the best-known ones, roughly in the order in which their history can be documented, and says where each came from. Where the origin is uncertain, it says so.

Counting numbers and the oldest patterns

The simplest sequence, 1, 2, 3, 4 and onward, is older than writing. Tallies on bones such as the Ishango bone from Central Africa show people marking counts long ago, and the prehistoric counting that preceded written digits likely included early ideas of odd and even. Exactly how much arithmetic those marks represent is debated, so it is wise to speak cautiously about what the makers intended.

From these come the first named patterns: the even numbers, the odd numbers and the powers of two. Powers of two, 1, 2, 4, 8, 16, arise from repeated doubling, and Egyptian multiplication relied on doubling. They took on a new role with binary notation, which Gottfried Leibniz described in the early eighteenth century and which modern computers use.

Figurate numbers: triangles and squares

The Greeks arranged pebbles into shapes. The triangular numbers, 1, 3, 6, 10, 15, count dots in a triangle, and the square numbers, 1, 4, 9, 16, 25, count dots in a square. Writers in the Pythagorean tradition valued these patterns, and later authors such as Nicomachus gave them a systematic treatment. The sequences are easy to describe and give a first taste of how diagrams can prove facts about numbers, such as the sum of the first few odd numbers always being a square.

Primes

The prime numbers, 2, 3, 5, 7, 11, are those with no divisors other than 1 and themselves. Euclid, around 300 BCE, proved that the list never ends, and the sieve of Eratosthenes gave a way to list them. The sequence has no simple formula, and its irregular spacing is the source of continuing research. Many of the most famous unsolved problems in mathematics concern primes.

Perfect numbers and amicable pairs

Perfect numbers equal the sum of their proper divisors: 6, 28, 496, 8128. Euclid linked them to primes of a particular form, and Euler later proved that every even perfect number has that form. Whether an odd perfect number exists is not known. The Greek tradition also noted amicable numbers such as 220 and 284, which are not a sequence in the strict sense but a family of pairs, with later contributions from Thabit ibn Qurra and Euler. The rarity of these numbers led ancient writers to treat them symbolically, which was a belief about the numbers rather than a property they have. The perfect number six is the best-known example of that attitude.

The Fibonacci numbers

The sequence 1, 1, 2, 3, 5, 8, 13, 21, in which each term is the sum of the previous two, bears the name of Leonardo of Pisa, who posed a puzzle about rabbit pairs in the Liber Abaci of 1202. Earlier Indian writers on Sanskrit metre, including Virahanka, described the same recurrence when counting syllable patterns. The ratio of successive terms approaches the golden ratio, about 1.618, which is why the two topics are often discussed together. Many popular claims about the sequence in nature and art go well beyond the evidence.

Pascal’s triangle

Arrange the binomial coefficients in a triangle, each entry the sum of the two above it: 1; 1, 1; 1, 2, 1; 1, 3, 3, 1; 1, 4, 6, 4, 1. Blaise Pascal wrote a treatise on the triangle in the seventeenth century, and it carries his name in Europe. But the pattern is much older. Indian commentators on Pingala’s work on metre described a related arrangement, the mathematician Yang Hui in thirteenth-century China published the triangle and credited an earlier writer, Jia Xian, and Omar Khayyam is reported to have known the underlying method. The triangle is a good example of an idea that is named for a late user rather than the first. The Fibonacci numbers also appear in it, as sums along shallow diagonals.

The harmonic series

The harmonic series, 1 plus 1/2 plus 1/3 plus 1/4 and so on, has terms that shrink toward zero, yet the total grows without bound. Nicole Oresme, a fourteenth-century French scholar, is credited with an early proof of this by grouping terms: 1/3 plus 1/4 is more than 1/2, then the next four terms exceed 1/2, and so on. It is a classic case of intuition failing, since terms that become tiny seem as though they should add up to a finite total.

The Collatz sequence

Not all famous sequences are ancient. In 1937, Lothar Collatz posed a question about the following rule: start with any positive whole number; if it is even, halve it; if it is odd, triple it and add 1. Starting at 6 gives 6, 3, 10, 5, 16, 8, 4, 2, 1. The conjecture is that every starting number eventually reaches 1. Computers have checked an enormous range of starting values without a counterexample, but no proof exists, and the problem has been described as easy to state and extremely hard to settle. It is a modern reminder that simple rules can resist analysis.

Cataloguing the lists

By the twentieth century there were so many sequences that cataloguing them became a project in itself. Neil Sloane began collecting integer sequences in the 1960s, and the resulting reference, now an online encyclopedia, lets anyone look up a list of numbers to see where it occurs. Its existence shows how many ways a short list of integers can arise.

Sequences and symbolism

Number lore often borrows from these sequences. Writers in the tradition of Pythagorean number philosophy found meaning in triangular and perfect numbers, and later numerologists drew on the same material. It helps to keep two things apart: the mathematical facts, which can be checked, and the meanings that cultures attached to them, which are beliefs and vary between traditions. The articles that follow in this cluster take individual sequences and ideas and trace them in more detail.

A map of origins

The table makes the broader point that these sequences came from many places and purposes: religious poetry, practical geometry, trade arithmetic and modern recreation. What they share is that each was named after a person or school, while the pattern itself had often been noticed earlier by someone else.