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Zeckendorf Representation

Tool to apply / check the Zeckendorf theorem stipulating that any integer can be written in the form of sum of non consecutive Fibonacci numbers also called Zeckendorf representation.

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Zeckendorf Representation -

Tag(s) : Arithmetics

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Zeckendorf Representation

Zeckendorf Representation Calculator


Tool to apply / check the Zeckendorf theorem stipulating that any integer can be written in the form of sum of non consecutive Fibonacci numbers also called Zeckendorf representation.

Answers to Questions

What is the Zeckendorf theorem? (Definition)

Every natural integer $ n \in \mathbb {N} $ has a unique representation in the form of a sum of non-consecutive Fibonacci numbers. Its formula is written: $$ n = \sum_{i=0}^{k} \alpha_i F_{i} $$ with $ F_i $ the ith Fibonacci number, $ \alpha_i $ is a binary number $ 0 $ or $ 1 $ (a way to indicate that the number of Fibonacci is in the sum, or it is not) and $ \alpha_i \times \alpha_{i + 1} = 0 $ (a way to prevent 2 numbers consecutive Fibonacci).

This proprety is used in Fibonacci coding (a binary representation of any integer based on the values of $ \alpha_i $ in the formula above)

How to calculate a Zeckendorf representation?

Enter a value of a number $ N $ and dCode will do the calculation automatically.

Example: 10000 is the sum of $ 6765 + 2584 + 610 + 34 + 5 + 2 $, respectively the 20th, 18th, 15th, 9th, 5th and 3rd Fibonacci numbers

Algorithmically, dCode uses Binet's formula to obtain Fibonacci numbers close to a given number and subtracts them recursively until finding the Zeckendorf representation.

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