[seqfan] Re: Non intersecting Fibonacci sets

[M. F. Hasler]

I agree with what precedes.
 I think it would be interesting to list subsequent numbers
0,1 (or 1,2); 4,6; 7,11; 9,14; ...
following Eric's idea:
F1 = F(0,1)
F2 = F(4,6) (earliest Fib-like seq. disjoint with F1)
F3 = F(7,11) = earliest F-seq. disjoint with F1 U F2
(NB: F(7,9) would have 16 in common with F1 U F2)
F4 = F(9,14) = earliest F-seq. disjoint with F1 U F2 U F3
(NB: F(9,12) would have 21 occurring earlier)
etc.

PS: just got LEJ's mail : well spotted !
Yet sequence ...,4,6,7,11,9,14...
= interleaved col.1 and 2 of A126714
isn't in OEIS and could be there, IMHO.

--  Maximilian


On Sat, Oct 11, 2014 at 1:08 PM, Allan Wechsler <acwacw at gmail.com> wrote:
> All distinct Fibonacci-like sequences eventually separate forever; this is
> fairly easy to prove, since the ratio between consecutive elements
> converges rapidly to phi, regardless of the starting seed. I suspect that
> there is an upper limit on the number of elements two Fibonacci-like
> sequences can have in common, something like 3 or 4.
>
> On Sat, Oct 11, 2014 at 12:51 PM, Eric Angelini <Eric.Angelini at kntv.be>
> wrote:
>
>> 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.