A000010

A000010 is the OEIS sequence for the Fibonacci numbers. This sequence begins with 0 and 1, and each subsequent number is the sum of the two preceding ones. The sequence starts 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The definition is commonly expressed by the recurrence relation F(n) = F(n-1) + F(n-2) with initial conditions F(0) = 0 and F(1) = 1.

The Fibonacci numbers appear in various areas of mathematics and have connections to the golden ratio. They are named after Leonardo of Pisa, later known as Fibonacci, who introduced them to Western European literature in his 1202 book Liber Abaci, although the sequence had been described earlier in Indian mathematics.

Applications of the Fibonacci sequence extend to computer algorithms, such as the Fibonacci search technique and Fibonacci heaps. They are also observed in natural phenomena, including the branching patterns of trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the uncurling of a fern. The ratio of successive Fibonacci numbers approaches the golden ratio, approximately 1.618.