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I need to count the number of triangles formed in a grid of $n\times n$ with positive integer coordinates $(0..n)$. For example for $n = 1$ the answer is 4.

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$0$ isn't a positive integer! –  Qiaochu Yuan May 31 '12 at 6:03
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I think exact formula may be very hard. The problem is dealing with lines that pass through $3$ or more lattice points. This is sensitive to number-theoretic properties of $n$. –  André Nicolas May 31 '12 at 6:03
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Calculate it for some small values of $n$, and look it up in the Online Encyclopedia of Integer Sequences. –  Gerry Myerson May 31 '12 at 11:45
    
@Qiaochu, I have read that in France, zero is considered to be positive. –  Gerry Myerson May 31 '12 at 11:48
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