Negafibonacci

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[edit] Definition

In mathematics, negaFibonacci numbers are the negatively indexed elements of the Fibonacci sequence.

The negaFibonacci numbers are defined inductively by the recurrence relation:

  • F-1 = 1,
  • F-2 = -1,
  • Fn = F(n+2)−F(n+1).

They may be defined by the formula F−n = (−1)n+1Fn.

The first 10 negaFibonacci numbers are F-1 = 1, F-2 = -1, F-3 = 2, F-4 = -3, F-5 = 5, F-6 = -8, F-7 = 13, F-8 = -21, F-9 = 34, F-10 = -55.

[edit] Integer representation

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Any integer can be uniquely represented[citation needed] 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 for coding numbers using a binary representation. 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.