NOTE This is for internal use NOT for publication

If the sequence satisfies the 'Fibonacci relation' then X₂=X₀+X₁ and if it is also geometric with ratio t and the first term is a then the next two terms are at and at² so that

X2 = at2 = a + at,

and the ratio of the first two terms is a solution of the quadratic equation

t2 - t - 1 = 0

giving t = (1 ±Ö5)/2. Note that

(1 - Ö5)/2 =  1 (1 + Ö5)/2 =  1 f .

Hence the ratio of the first two terms must be 1 : f or f: 1.
Conversely, if the ratio X₁/X₀ of the first two terms is t where t² = 1 + t then X₁=tX₀ and X₂=X₁+X₀=t²X₀. We prove by induction that X_n=tⁿX₀ so that it is a geometric progression.
Suppose X_k=t^kX₀ for all k < n. Then

Xn = Xn-1+Xn-2 = [tn-1+tn-2]X0 = tn-2(t+1)X0=tnX0.