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Transcript

Fibonacci Numbers Fibonacci Numbers • • • • • • • In mathematics, the Fibonacci numbers are a sequence of numbers named after Leonardo of Pisa, known as Fibonacci, whose Liber Abaci published in 1202 introduced the sequence to Western European mathematics. The sequence is defined by the following recurrence relation: That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, … Each third number of the series is an even number. The sequence named after Fibonacci was first described in Indian mathematics. The sequence extended to negative index n satisfies Fn = Fn−1 + Fn−2 for all integers n, and F-n = (−1)n+1Fn: .., -8, 5, -3, 2, -1, 1, followed by the sequence above. Fibonacci Rectangles 1 1 1 1 1 2 1 1 2 3 1 1 2 3 5 1 1 8 2 3 5 13 1 1 8 2 3 5 13 1 1 8 2 3 5