A006906 question

[Paul D. Hanna]

Dean (and Seqfnas), 
     Good work.  Yet there may yet be a limit here that does exist.  
Is it not accurate to state: 
 
(*) the limit of a(n+3)/a(n) exists and is equal to 3. 
 
This can still be true and the limit of (*) unique 
even though a(n+1)/a(n) takes on 3 different values -  
as long as the product of these 3 limit points equals 3. 
 
Does that (*) check out as true in your analysis? 
 
Empirically, in my trials, (*) seems to hold very well. 
Thanks, 
     Paul 
 
> > For A006906, could a(n+1)/a(n) be approaching a limit?
> 
> > Like maybe 3^(1/3)?
> 
> The limit does not exist.  The generating function for a(n) is
[...] 
> Because these 3 values are different, the sequence of ratios 
> a(n+1)/a(n) has
> 3 different limit points, which differ slightly from 3^(1/3).