R_r recurrence even simpler than S_r

[Ralf Stephan]

> Am 04.04.04 01:39 schrieb Rainer Rosenthal:
> > We may consider (1) as the special case of 
> > 
> >       a(n-1)*a(n+1) = a(n)^2 - 1         (2)
> > 
> > where a(n) = n = 0, 1, 2, ...
> > 
> > Looking for other sequences 0, 1, r, ... obeying (2) we
> > quickly find many of them already in the OEIS:
> > 
> > r= 2: 0,1,2,3,4,5,6,...      "the numbers"
> > r= 3: 0,1,3,8,21,55,144,...  A001906
> > r= 4: 0,1,4,15,56,209,...    A001353
> > r= 5: 0,1,5,24,115,551,...   A004254
> > r= 6: 0,1,6,35,204,1189,...  A001109
> > 
> > followed by A004187, A001090, A018913, A004189, A004190,
> > A004191, A078362, A007655, A078364, A077412, A078366,
> > A049660, A078368, A075843 (r=7 up to r=20).

just to add the (conjectured) g.f.

   V(r,z) = 1/(1-rz+z^2).

If true it means you can get from R_k to S_{k+2} by doing partial sums
followed by pairwise sums and a shift! Now what would be the reverse 
transformation in the fewest words?


ralf