![]() Lekkerkerker had published the same result 20 years earlier. This is known as Zeckendorf’s theorem, published by E. Programming exercise: How would you implement a function that finds the largest Fibonacci number less than or equal to its input? Once you have this it’s easy to write a program to find Fibonacci representations. You can find the Fibonacci representation of a number x using a greedy algorithm: Subtract the largest Fibonacci number from x that you can, then subtract the largest Fibonacci number you can from the remainder, etc. So you could think of a Fibonacci sum representation for x as roughly a base φ representation for √5 x. The nth Fibonacci number is approximately φ n/√5 where φ = 1.618… is the golden ratio. ![]() In the example above, 8 = 5 + 3 and so you could write 10 as 5 + 3 + 2. This decomposition is unique if you impose the extra requirement that consecutive Fibonacci numbers are not allowed. It’s easy to see that the rule against consecutive Fibonacci numbers is necessary for uniqueness. It’s not as easy to see that the rule is sufficient.Įvery Fibonacci number is itself the sum of two consecutive Fibonacci numbers-that’s how they’re defined-so clearly there are at least two ways to write a Fibonacci number as the sum of Fibonacci numbers, either just itself or its two predecessors. ![]() For example, 10 = 8 + 2, the sum of the fifth Fibonacci number and the second. For example, almost 2500 years ago, a Greek sculptor and architect named Phidias. Every positive integer can be written as the sum of distinct Fibonacci numbers. But the numbers in Fibonaccis sequence have a life far beyond rabbits. List of Fibonacci numbers In mathematics, the Fibonacci numbers form a sequence such that each number is the sum of the two preceding numbers, starting from 0 and 1. ![]()
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