Understanding the Fibonacci Sequence
The Fibonacci sequence is defined by a simple recurrence: each number is the sum of the two before it. Formally, F_n = F_(n-1) + F_(n-2), starting from F_0 = 0 and F_1 = 1. This produces the familiar sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... continuing indefinitely.
The sequence was introduced to Western mathematics by Leonardo of Pisa ("Fibonacci") in his 1202 book Liber Abaci, using a thought experiment about idealized rabbit population growth. It shows up in phyllotaxis (leaf and seed arrangement patterns), in computer science algorithm design, and in the way it connects to the Golden Ratio described below.
As you divide consecutive Fibonacci numbers (F_n / F_(n-1)) for larger and larger n, the result converges toward the irrational constant known as the Golden Ratio, φ ≈ 1.618034.