The sequence behind spirals and rabbits
Start with 0 and 1, then keep adding the last two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… This is the Fibonacci sequence, named after Leonardo of Pisa, who used it in his 1202 book Liber Abaci in a puzzle about how fast rabbits breed. Indian mathematicians had described it centuries earlier in the study of poetic meters. The sequence appears in surprising places, from the spirals of sunflower seeds and pine cones to algorithms and financial charts.
Definition
F(0) = 0, F(1) = 1, and F(n) = F(n − 1) + F(n − 2)
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| F(n) | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 |
Binet’s formula
Computing F(100) by adding term after term is tedious. Binet’s formula, published by Jacques Binet in 1843 (and known earlier to de Moivre), gives any term directly:
F(n) = (φⁿ − ψⁿ) ÷ √5, where φ = (1 + √5) ÷ 2 ≈ 1.6180339887 is the golden ratio and ψ = (1 − √5) ÷ 2 ≈ −0.618.
Because |ψ| is less than 1, ψⁿ quickly becomes negligible, so F(n) is simply φⁿ ÷ √5 rounded to the nearest whole number. The calculator uses this and works up to n = 78, the largest term it can display exactly.
Worked example
For n = 20:
- F(20) = φ²⁰ ÷ √5 rounded = 6,765.
- Previous term F(19) = 4,181; next term F(21) = 10,946.
- Sum of the first 20 terms: F(22) − 1 = 17,710.
- Ratio F(20) ÷ F(19) = 1.618033963, already within 0.00000003 of φ.
- Lucas number L(20) = 15,127.
Useful properties
- Sum: F(1) + F(2) + … + F(n) = F(n + 2) − 1.
- Ratio: F(n + 1) ÷ F(n) tends to φ as n grows.
- Test: a whole number x is a Fibonacci number exactly when 5x² + 4 or 5x² − 4 is a perfect square. For 144, 5 × 144² + 4 = 103,684 = 322², so 144 is F(12).
- Divisibility: F(m) divides F(n) whenever m divides n; for example F(3) = 2 divides every third term.
- Lucas numbers: 2, 1, 3, 4, 7, 11, 18… follow the same rule with a different start, and L(n) = φⁿ + ψⁿ.
In nature and elsewhere
Many plants arrange leaves and seeds at angles related to the golden ratio, so counting the spirals on a sunflower head or a pine cone often gives consecutive Fibonacci numbers such as 34 and 55. In computer science, Fibonacci numbers appear in the analysis of algorithms and data structures such as Fibonacci heaps. Traders use “Fibonacci retracements”, though there is little evidence that markets respect them.