The Fibonacci sequence explained

The rule and the first terms

Like Kaprekar’s routine, the sequence is produced by repeating one simple step. The difference is that Kaprekar’s routine works on a finite set of four-digit numbers and must settle down, while the Fibonacci numbers grow without limit.

Where the sequence comes from

The name comes from Leonardo of Pisa, later called Fibonacci, whose 1202 book Liber Abaci helped bring Hindu–Arabic numerals to Europe. One exercise in the book asks how many pairs of rabbits there will be after a year if every mature pair produces a new pair each month. The monthly totals follow the sequence.

The numbers were known much earlier in India. Scholars of Sanskrit poetry, including Virahanka and later Hemachandra, counted the ways to fill a line of verse with short and long syllables. The number of arrangements follows the same rule, because a line of length n ends either with a short syllable after a line of length n − 1 or with a long syllable after a line of length n − 2.

Ratios that approach the golden ratio

Divide each term by the one before it:

The ratios approach φ = (1 + √5) / 2 ≈ 1.6180339887, the golden ratio. The approach alternates: one ratio is slightly too large, the next slightly too small. This is a consequence of Binet’s formula, which gives any Fibonacci number directly:

Because |ψ| is less than 1, the ψⁿ term shrinks quickly, and F(n) is simply φⁿ / √5 rounded to the nearest whole number. More on φ, including the popular myths about it, is in the golden ratio: facts and myths.

Surprising patterns

• Every third Fibonacci number is even: 0, 2, 8, 34, 144, … The pattern of odd and even repeats odd, odd, even.
• Cassini’s identity: F(n − 1) · F(n + 1) − F(n)² = (−1)ⁿ. For example 5 · 13 − 8² = 1 and 8 · 21 − 13² = −1.
• The sum of the first n terms is one less than a later term: 1 + 1 + 2 + 3 + 5 + 8 = 20 = 21 − 1.
• F(m) divides F(n) whenever m divides n. For example F(5) = 5 divides F(10) = 55.

The last digits repeat every 60 terms

Look only at the last digit of each Fibonacci number: 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, … Because each last digit depends only on the two previous last digits, and there are only 100 possible pairs, the pattern must eventually repeat. It does so after exactly 60 terms. This repeat length is called the Pisano period.

For other divisors the periods differ: modulo 2 the period is 3, modulo 3 it is 8, and modulo 5 it is 20. The argument is the same finite-state reasoning that guarantees Kaprekar’s routine must end in a cycle.

Every number is a sum of Fibonacci numbers

Zeckendorf’s theorem says every positive whole number can be written in exactly one way as a sum of Fibonacci numbers that are not neighbours in the sequence. Take the largest Fibonacci number that fits and repeat with the remainder:

A practical by-product: because consecutive Fibonacci ratios are close to 1.609, the number of kilometres in a mile, you can convert roughly by moving one step along the sequence. 5 miles ≈ 8 km, 8 miles ≈ 13 km and 13 miles ≈ 21 km.

Fibonacci in nature: what is true

In many plants the number of visible spirals in a sunflower head, pine cone or pineapple is a Fibonacci number, and neighbouring spiral families often come in consecutive pairs such as 34 and 55 or 55 and 89. The usual explanation is that new seeds or scales grow at a roughly constant angle of about 137.5°, the golden angle, which packs them efficiently. This produces Fibonacci spiral counts as a mathematical consequence.

It is not a universal law. Studies of large numbers of sunflowers have found many heads with non-Fibonacci counts, and many plants show no such pattern. Claims that Fibonacci numbers govern galaxies, hurricanes or human beauty are not supported by evidence. A similar caution applies to 6174 in nature.