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

[Rainer Rosenthal]

> Let me preserve brevity by not answering this stumper:
> What's next in 45 96 175 288 320 640 891 1200 1350?
> (Hint: it has something to do with Rainer's S_r sequences.)

I guess you meant 45 96 175 288 441 640 891 1200 1573 2016

       a(n) = S_n(4)

But may be, I am wrong, since there is 
http://www.research.att.com/projects/OEIS?Anum=A028687
(Sorted k-factorial numbers (numbers of form k-1 excluded)

which is the same as S_n(4), if you cancel the elements with
index 3, 8, 9, 17, 25, 28, 38, 40 (and add a weird 0 :-)

A028687: 
1,3,16,21,45,96,175,288,315,416,441,640,891,1200,1573,2016,2535

S_n(4):
1,0,3,16,21,45,96,175,288,441,640,891,1200,1573,2016,2535


Thanks to Benoit Cloitre, who provided some background for
my observations (Somos sequences and Laurent phenomen)
I will open a thread in sci.math for that, since we are
leaving the OEIS related stuff now. It all started with S_36,
emerging from a thread in alt.math.recreational
Message-ID: <DLz2c.47129$kR4.1466535 at weber.videotron.net>
with convergents of the c.f. of sqrt(2) and gave interesting
relashionships between existing OEIS sequences. Finally we
reached at the missing S_6, which is related to A002531
and the convergents of the c.f. of sqrt(3). 

  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   WOW!  A tour de force from sqrt(2) to sqrt(3)  :-)))
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Thanks to Henry in Rotherhithe, who showed S_1, S_2 and S_3.
I did not find them, because a(n+1) = (a(n)-1)^2/a(n-1) gets
a division by zero error :-)

Wishing a happy sunday
Rainer Rosenthal
r.rosenthal at web.de