Re: Defining the Fibonacci numbers with only one starting value

> But, thinking this over, I've concluded that a modification
> of this alternate definition of the FNs actually allows
> using only _one_ starting number, as in
>
> F(3) = 2;
> N > 0 => F(2*N-1) = F(N)^2 + F(N-1)^2 and
> F(2*N) = F(N+1)^2 - F(N-1)^2.
>
> For N = 2, this yields F(3) = 2 = F(2)^2 + F(1)^2,
> which can be true only if F(2) = F(1) = 1.

Well, you'll need the assumption that F(n)'s are non-negative integers.

> And then for N = 1, it yields F(2) = F(2)^2 - F(0)^2,
> which, for F(2) = 1, can be true only F(0) = 0
> (or, F(1) = F(1)^2 + F(0)^2, again true for
> F(1) = 1 only if F(0) = 0).

Actually few people define F_n to be started from 0. Most of the definition
starts from 1.

> Interesting, defining a series in which each member except
> one depends on two others, starting with just one member.

With an extra assumption.


.