Fibonacci

Representation of numbers (part 2)

We now know that each positive integer can be written as a sum of different Fibonacci numbers.
The numbers F₁ and F₂ were considered as being different, although they have the same numerical value 1.
This leads us to the following question: Suppose we cancel one element from the sequence of Fibonacci numbers (e.g. the number F₂). Is it possible to represent each positive number as a sum of the remaining Fibonacci numbers? This question will now be answered. We assume that F_n has been canceled (n>1).
If for certain k>n each number less than F_k has a representation without term F_n, then that will also be true for each number less than F_k+1 (Add F_k to the representations for 0,1,2,...,F_k-1-1), and therefore also for all numbers less than F_k+2 et cetera.
As we already know, all numbers less than F_n+1 are representable with different terms from F₁,...,F_n-1. So, we may choose k=n+1. This completes the proofs.

Deleting two terms in the Fibonacciseries is not allowed, because if we delete e.g. F_a and F_n with a<n, then (F₁+F₂+...+F_n) - F_n - F_a = F_n+2 - 1 - F_n - F_a = F_n+1 - 1 - F_a < F_n+1-1. So F_n+1-1 is not representable.