[seqfan] Non intersecting Fibonacci sets

[Eric Angelini]

Hello SeqFans,
I guess this is old hat, sorry;
Let's name Fi(1,2) the Fibonacci-like
sequence starting with 1 and 2;
Is there a Fi(a,b) seq that has no term
in common with Fi(1,2)?
If yes, what is the lexico-first such seq?
Could it be Fi(4,6)?
If Fi(1,2) shares no term with Fi(4,6),
is there a Fi(x,y)  seq that shares no term
with Fi(1,2) AND Fi(4,6)?
If yes, can we iterate that construction
and compute more and more Fi(p,q) such seq?
We might, in this way, arrange all integers >0
in non intersecting Fi-sets and compute the seq S 
of the successive smallest integers belonging to each Fi-set.
Best,
É.