Every positive integer n can be represented as n = Sum_{k>=0} (b(i)*F(2*k)), where b(i) in {-1, 0, 1} , F(n) – Fibonacci sequence. Trivial proof follows from the Zeckendorf’s theorem. Having a Zeckendorf representation of any positive integer, you can also present this integer by sum or difference of some Fibonacci sequence elements with a even index. Example: We have such a Zeckendorf representation of the number 100=89+8+3 where 89 is an element with an odd index, it can be replaced by the difference of two neighboring elements 89=144-55, for the case when in the Zeckendorf representation of the integer are followed by several neighboard elements of Fibonacci sequence with an odd index, they can also be replaced: Sum_{k..n} F(2k-1) = F(2n) – F(2k-2) where k, n depends on the Zeckendorf representation. Example: 18=13+5, k=3, n=4, 18=21-3. – Zhandos Mambetaliyev, Mar 05 2020
Jandos Mambetali 