Definition and the rabbit problem
The Fibonacci sequence starts with two 1s (or with 0 and 1), and every later term is the sum of the previous two. It is named after Leonardo of Pisa, known as Fibonacci, who included a puzzle about rabbits in his 1202 book Liber Abaci: if a newborn pair of rabbits starts breeding after one month and then produces a new pair every month, how many pairs are there after a year? Month by month the answer runs 1, 1, 2, 3, 5, 8 and so on. The numbers were known much earlier to Indian prosodists, who used them to count patterns of long and short syllables. The name "Fibonacci sequence" was popularised by the French mathematician Édouard Lucas in the 19th century.
Ratios that converge on the golden ratio
Divide each term by the one before and you get 2, 1.5, 1.667, 1.6, 1.625…, swinging above and below the golden ratio φ = (1+√5)/2 ≈ 1.6180 and closing in on it. That link gives Binet's formula, which computes the nth term directly from powers of φ. The sequence has many other neat properties, such as Zeckendorf's theorem: every positive integer can be written in exactly one way as a sum of non-consecutive Fibonacci numbers.
| n | F(n) | F(n)/F(n−1) |
|---|---|---|
| 1 | 1 | — |
| 2 | 1 | 1.000 |
| 3 | 2 | 2.000 |
| 4 | 3 | 1.500 |
| 5 | 5 | 1.667 |
| 6 | 8 | 1.600 |
| 7 | 13 | 1.625 |
| 8 | 21 | 1.615 |
| 9 | 34 | 1.619 |
| 10 | 55 | 1.6176 |
| 11 | 89 | 1.6182 |
| 12 | 144 | 1.6180 |
Fibonacci in nature and daily life
Sunflower seeds form two sets of spirals winding in opposite directions, and the counts are often neighbouring Fibonacci numbers such as 34 and 55 or 55 and 89; pine cones (8 and 13) and pineapples show the same pattern. Experiments and models suggest the reason: placing each new leaf or seed about 137.5 degrees (the golden angle) from the last packs them most evenly. Not every plant follows the rule, though, and the often-cited nautilus shell is not a golden spiral. In trading, Fibonacci retracement levels (23.6%, 38.2%, 61.8%) are popular chart guides, but there is little scientific evidence that prices reliably turn at them. In computer science the sequence is the classic first example of recursion and dynamic programming.
FAQ
- Does the Fibonacci sequence start at 0 or 1?
- Both conventions are used. Mathematicians often define F(0)=0 and F(1)=1, while the original rabbit problem starts 1, 1. The later terms are the same either way.
- Does nature really follow the Fibonacci sequence?
- It appears often in plants whose leaves or seeds grow in spirals, such as sunflowers and pine cones, but not everywhere. Some popular examples are exaggerated or cherry-picked.
Sources reviewed: 2026-10-26
References
- Sigler, L. E. (2002). Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation. Springer.
- Singh, P. (1985). The so-called Fibonacci numbers in ancient and medieval India. Historia Mathematica, 12(3), 229–244.
- Koshy, T. (2018). Fibonacci and Lucas Numbers with Applications (2nd ed.). Wiley.
- Douady, S., & Couder, Y. (1992). Phyllotaxis as a physical self-organized growth process. Physical Review Letters, 68(13), 2098–2101.