Does there exist a formula for counting the number of lattice points not outside of a square , with the at most information available concerning the square are the position coordinates of the four corners of the square ?
Number of lattice points
3
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geometry
integer-lattices
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0@ $B$obaFret:- I have looked over integer (of-course positive) values of n , there the formula seems to hold – 2012-10-19
2 Answers
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Consider the square with vertices $(2,0), (4,2), (2,4), $ and $(0,2)$, then I think there are $13 $ points not lying outside the square. But here $n=2\sqrt {2}$ and $(1+[n])^2=9.$
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0@SouvikDey: Yes I calculated by hand, but I think the no. of lattice points can be calculated depending on the angle of rotation of the square about it's centre. – 2012-10-19
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What about the square with corners $(\pm1/2,\pm1/2)$? Here $n=1$ is an integer, but there's only one lattice point, the origin, not outside the square, not 4.
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1Well, if you know the corners, you might be able to find $b$. For example, suppose $n=5$, and there's a corner at $(0,0)$. If another corner is at $(5,0)$, then $b=20$, but if another corner is $(3,4)$, then $b=4$. – 2012-10-19