Tool for encoding/decoding numbers using Fibonacci encoding (binary words never having two consecutive 1 values)

Fibonacci Encoding - dCode

Tag(s) : Compression, Mathematics

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The Fibonacci code uses the Zeckendorf theorem (and Zeckendorf's representation of a number) which states that any integer can be written as the sum of non-consecutive Fibonacci numbers.

$$ n = \sum_{i=1}^{k} \beta_i F_{i} $$

The Fibonnacci coding consists in noting the coefficients $ \beta_i $ (being `0` or `1`) to make a binary number.

__Example:__ $ 123 $ is the sum of $ F_{11} = 89 $ and $ F_{9} = 34 $ or `1010000000` in binary (the two `1` are in position `8` and `10` starting from the right).

As the Zeckendorf representation never has 2 consecutive Fibonnacci numbers, the binary value will never have 2 times the number `1` consecutively.

Each `1` of the binary word corresponds to a Fibonacci number, to find the decimal number, add all the Fibonacci numbers corresponding to the `1` of the binary word.

__Example:__ `10100` corresponds to $ 1 \times F_5 + 0 \times F_4 + 0 \times F_3 + 1 \times F_2 + 0 \times F_1 = F_5 + F_3 = 8 + 3 = 11 $

A variant of Zeckendorf's theorem indicates that it is also possible to write any integer as the sum of non-consecutive Nega-Fibonacci (Generalization of Fibonnacci with negative indices) numbers, this encoding is called NegaFibonacci encoding.

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