R_r recurrence even simpler than S_r

[Rainer Rosenthal]

Dear SeqFan,

for the sequence a(n) = n^2 we have the recurrence
a(0)=0 a(1)=1 a(2)=4  and  a(n-1)*a(n+1)=(a(n)-1)^2.
We get a lot of interesting sequences by varying 
a(2)=r. These are listed partly in the comments of
sequence A001110 (the squaretriangles), which happens
to be S_36.
I like to thank all of you, who made helpful comments
some weeks ago, when we discussed this here.

     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
     And here is a new and similiar observation:

Contemplating S_4 = "the squares", you will notice
that the recurrence above is (n-1)^2*(n+1)^2=(n^2-1)^2.
This is an easy consequence of 

        (n-1)(n+1) = n^2 - 1             (1)

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).

Let me call these sequences "R_r type sequences". R_2 is
"the numbers" and R_3 up to R_20 are in the OEIS already.
The first R_r type sequence not in the OEIS is R_21:
R_21 = 0, 1, 21, 440, 9219, 193159, 4047120, ...
But this sequence is strongly related to A041833, which
gives the denominators in the c.f. of sqrt(437).

Many of these sequences are provided by Wolfdieter Lang
and characterized as members of a so called "m-family". 
Sequence R_22 is his A077421, but it lacks the corresponding
remark. (There are relations to A041219 and A041917).

If I haven't overlooked something then there are many 
explicit recurrences given in the cited sequences, but none
of them shows that most simple recurrence (2).

( I promised to make comments for the S_r type sequences and
will do so in the next weeks. Now it seems as if I should
add comments for R_r type sequences too. OK?)

Best regards,
Rainer Rosenthal
r.rosenthal at web.de