Pick's theorem

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Given a simply closed lattice polygon with area $\scriptstyle A\,$, when we have $\scriptstyle E\,$ counting the number of lattice points on the polygon edges and $\scriptstyle I\,$ counting the number of points in the interior of the polygon, the area can be determined with the following equation:

$A=I+{\frac {1}{2}}E-1.\,$

In the case of a lattice right triangle with legs $3n$ and $4n$, and hypotenuse $5n$, Pick's theorem gives us the formula $6n^{2}-4n+1$ for determining the number of integer lattice points inside the triangle (see A126587).

A126587 a(n) = number of integer lattice points inside the right-angle triangle with legs 3n and 4n (and hypotenuse 5n).

{3, 17, 43, 81, 131, 193, 267, 353, 451, 561, 683, 817, 963, 1121, 1291, 1473, 1667, 1873, 2091, 2321, 2563, 2817, 3083, 3361, 3651, 3953, 4267, 4593, 4931, 5281, 5643, 6017, 6403, ...}

Generalization

Using Ehrhart polynomials, this formula has been generalized to dimension three and higher.

External links

• Tom Davis, Pick's Theorem, 2003.
• Weisstein, Eric W., Pick's Theorem, from MathWorld—A Wolfram Web Resource..
• Weisstein, Eric W., Ehrhart Polynomial, from MathWorld—A Wolfram Web Resource..