Okay, here's one for people who are "math guys":

I have a sequence such that

x_1 = 1

and

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

for n>=1.

So the sequence begins 1, 2, 3, 5, 8, ...

Hypothesis: all x_n are integers.

Puzzle: prove or disprove the hypothesis.

This one has been bugging me on and off for over 10 years, so if someone has a solution, I'll be happy all day.