Representation With Negafibonacci Numbers
The Fibonacci sequence can be extended to negative index n using the re-arranged recurrence relation
which yields the sequence of "negafibonacci" numbers satisfying
Any integer can be uniquely represented as a sum of negafibonacci numbers in which no two consecutive negafibonacci numbers are used. For example:
- −11 = F−4 + F−6 = (−3) + (−8)
- 12 = F−2 + F−7 = (−1) + 13
- 24 = F−1 + F−4 + F−6 + F−9 = 1 + (−3) + (−8) + 34
- −43 = F−2 + F−7 + F−10 = (−1) + 13 + (−55)
- 0 is represented by the empty sum.
Note that 0 = F−1 + F−2 , for example, so the uniqueness of the representation does depend on the condition that no two consecutive negafibonacci numbers are used.
This gives a system of coding integers, similar to the representation of Zeckendorf's theorem. In the string representing the integer x, the nth digit is 1 if Fn appears in the sum that represents x; that digit is 0 otherwise. For example, 24 may be represented by the string 100101001, which has the digit 1 in places 9, 6, 4, and 1, because 24 = F−1 + F−4 + F−6 + F−9 . The integer x is represented by a string of odd length if and only if x > 0.
Read more about this topic: Zeckendorf's Theorem
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