What next in 0,1,6,25,96,361,1350,5041?

[Rainer Rosenthal]

Another way of putting the question above:

What have the following sequences in common:

S_4   = A000290    the squares
S_5   = A004146    Alternate Lucas Numbers - 2
S_7   = A054493    A Pellian-related sequence
S_8   = A001108    a(n)-th triangular number is a square
S_9   = A049684    F(2n)^2 where F() = Fibonacci numbers
S_20  = A049683    a(n)=(L(6n)-2)/16, L=Lucas Sequence
S_25  = A089927*   Expansion of 1/((1-x^2)(1-5x+x^2))
S_36  = A001110    Both triangular and square
S_49  = A049682    a(n)=(L(8n)-2)/45, L=Lucas sequence
S_144 = A004191^2  a(n)=S(n,12) (Chebyshev's poly 2. kind)

where A089927*(n) = A089927(2n-2)   


Answer: for any three successive members a, b, c we have
        a * c = (b-1)^2, i.e. the obey the recurrence

     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        a(n-1) * a(n+1) = ( a(n) - 1 )^2
     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

All S_r sequences start with 0, 1, r.

I will enter sequence S_6 in the next few days, which is
the first S_r not in the OEIS.

But A002531 is closely related to S_6, as can be seen from 
the following table:

     S_6     A002531^2   2*A002531^2-2
----------------------------------------
       0                        0
       1           1
       6                        6
      25          25
      96                       96
     361         361
    1350                     1350
    5041        5041
   18816                    18816
   70225       70225
  262086                   262086
         
I think it is a fine observation that so many sequences in
the OEIS obey the same recurrence rule  a c = (b-1)^2.
And funny enough this rule is never mentioned for any of
these sequences! Shall I add this as a remark for them?

I recall that this recurrence has been stated some days ago
by "zaphod" alias Colin Dickson for the squaretriangles, i.e.
for sequence A001110.

Let me finish my lengthy posting with the answer to the
question in the subject line: 18816 (see table).

Best regards
Rainer Rosenthal
r.rosenthal at web.de