Trying variants of a simple mathematical rule that yields interesting results can lead to additional discoveries and curiosities.
The numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55 belong to a famous sequence named for the Italian mathematician Fibonacci, who lived more than 700 years ago. Each consecutive number is the sum of the two numbers that precede it.
Suppose the first two numbers are instead 2 and 1. Applying the same addition rule produces a new sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, and so on. This string of numbers is the Lucas sequence, honoring the 19th-century French mathematician Édouard Lucas (1842–1891), who studied the Fibonacci sequence (and gave it its name). Lucas worked out what would happen if you started with any two whole numbers, then followed the Fibonacci rule.