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 $n$:
Show that: $t_{n-1}+t_n=n^2$
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elementary-number-theory
algorithms
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0You 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
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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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0This is a proof? – 2012-12-01