A000045

A000045 is the integer sequence known as the Fibonacci numbers. It begins with 0 and 1, and each subsequent number is the sum of the two preceding ones. The sequence is defined by the recurrence relation F(n) = F(n-1) + F(n-2), with initial conditions F(0) = 0 and F(1) = 1. The first few terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

The Fibonacci sequence has numerous applications in mathematics, computer science, and nature. It appears in the branching patterns of trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the uncurling of a fern, and the flowering of an artichoke. It is also closely related to the golden ratio, approximately 1.618. As n approaches infinity, the ratio of consecutive Fibonacci numbers F(n)/F(n-1) approaches the golden ratio. This mathematical constant has been observed in art, architecture, and design for centuries due to its aesthetically pleasing properties. The sequence has also found use in algorithms for searching and sorting data, and in financial market analysis.