My son Dan has been very interested in Fibonacci numbers recently.
He posits the following:
Given the sequence of Fibonacci numbers*, in which Fx represents the
xth term in the sequence, then, for all odd values of n:
1. all non-negative integers from Fn-1 through f(n+2)-2
(inclusive)
may be expressed as the sum of no more than n-2 Fibbonacci numbers.
2. The number F(n+2)-2 cannot be expressed as the sum of fewer
than n-1 Fibbonacci numbers.
I believe that (1) can be proven by induction. I don't know how to
go about proving/testing (2).
_________________________________________________________________________
* i.e. 1, 1, 2, 3, 5, 8, 13, 21, 34...
|
