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How to can prove that : $ t_{n-1}+t_n=n^2.$ where $t_n$ is number of points with integers coordinates in a square isosceles triangle of side $n$:

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    You probably need the square to have one vertex at a point with integer coordinate and sides parallel to the axes.2012-12-01

2 Answers 2

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You can use the expression:

$ t_n=\frac{n(n+1)}{2} $

This comes from summing the finite series

$ \sum_{k=1}^n k, $ The $k$ is the number of lattice points in each row of your triangle, and there are $n$ rows.

So,

$ t_n+t_{n-1}=\frac{n(n+1)}{2}+\frac{(n-1)n}{2}=\frac{n^2+n+n^2-n}{2}=n^2 $

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Because the dots happen to be arranged in a $n\times n$ square.

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    This is a proof?2012-12-01