In reply to:

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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

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Yes, I do really mean "/n", and yes, the numbers do get very big very quickly - it doesn't take long to run out of screen space And sorry for the mistake in my mental arithmetic - x_5 is indeed 10.

It's a long time since I've actually looked at this one, but ISTR making some progress by extending the sequence to include x_0 = 1. Then the expression becomes

x_{n+1} = sum_i=0^n{{x_i}^2} / n

and

n.x_{n+1} = (n-1)x_n + {x_n}^2

n (x_{n+1} - x_n} = {x_n}^2 - x_n
= x_n (x_n - 1)

So

x_{n+1} = x_n + x_n(x_n - 1) / n

Which would be great if I could somehow show that x_n or x_n - 1 is a multiple of n ... Unfortunately, it appears to alternate, so that for odd n, x_n is a multiple of n, and for even n, x_n - 1 is a multiple of n.

(Hmm, I'd forgotten how ugly TeX notation can be when you're not used to it)