so if your sequence is correct, then I think you mean "/ x_n" rather than "/ n". If you mean "/ n" then I think the sequence is 10 not 8, and after about n = 8 or so the numbers just explode upwards -- so much so that I can't even check to see if they really are integers (i wanted to run a few rounds to make sure the hypothesis was right).
However, if you meant "/ x_n" then that's a whole different ballgame... And those provably are integers, and it's actually the fibonacci sequence ... here's why:
define sum(a(b), b, c, d) to be "the sum of a(b) where b goes from c to d" .. to make parsing easier.
i'm switching to definiing n in terms of n-1 rahter than n+1 in terms of n... it made my brain hurt the other way for some reason.
restatement:
x_n = ( 1 + sum( (x_i)^2, i, 1, n-1) ) / x_(n-1)
pull out the last value of the sum:
x_n = ( 1 + sum( (x_i)^2, i, 1, n - 2) + (x_(n-1))^2 ) / x_(n-1)
divide through:
x_n = x_(n-1) + (1 + sum( (x_i)^2, i, 1, n - 2)) / x_(n-1)
the 1 + sum(.., n - 2) by definition is actually x_(n-1)*x_(n-2), which makes it now:
x_n = x_(n-1) + (x_(n-1)*x_(n-2)) / x_(n-1)
simplify and you get:
x_n = x_(n-1) + x_(n-2)
Like I mentioned, though, if you meant "/ n", then I need to go back to the drawing board.
Mike