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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.

### 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.

## Source code

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