03 — The Sequence

1, 1, 2, 3, 5, 8…

Fₙ = Fₙ₋₁ + Fₙ₋₂  —  each term the sum of the two before it

The rule could not be simpler: start with 1 and 1, and every number after is the sum of the two before it. 1, 1, 2, 3, 5, 8, 13, 21, and onward, forever. It is named for Leonardo of Pisa, who introduced it to Europe in 1202 through a hypothetical problem about breeding rabbits in his book Liber Abaci. Within a few generations the name stuck.

But the sequence has nothing to do with rabbits, and it is centuries older than Fibonacci. Indian scholars had already worked it out, in an entirely different discipline: the study of Sanskrit poetic meter. Counting the number of ways to fill a fixed-length rhythm using only short, one-unit syllables and long, two-unit syllables, the grammarian Pingala touched on the pattern around the 2nd century BCE. Virahanka gave it a clean recursive rule around the 7th century CE. By the 12th century the poet-monk Hemachandra had written it out in full — decades before Fibonacci's book reached Europe. The numbers were identical either way. Only the reason for counting them differed: rabbits in one telling, syllables in the other.

As the sequence grows, the ratio between consecutive terms — 5/3, 8/5, 13/8, 21/13 — closes in on a single irrational number: φ, the golden ratio, the same constant behind the golden angle.

Fₙ = (φₙ − ψₙ) ⁄ √5

where ψ = (1 − √5)⁄2. Even though the sequence is built from nothing but whole-number addition, its exact value at any position secretly depends on an irrational number. The formula is usually credited to Binet, who published it in 1843 — though Abraham de Moivre had already found it more than a century earlier, and Daniel Bernoulli and Euler both knew of it too. Even Binet's claim to it was a rediscovery.

The sequence turns up in places with no obvious connection to poetry or to plants. A honeybee drone hatches from an unfertilized egg, so it has exactly one parent — a mother, no father. His mother, a fertilized queen, had two parents. Trace the drone's family tree backward, one generation at a time, and the count of ancestors runs 1, 1, 2, 3, 5, 8, 13 — the sequence again, arrived at this time by nothing more than which bees happen to need a father and which don't.

A rule simple enough for a child, reached independently by poets, kings, and insects that never once compared notes.