Sticks and Triangles

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
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Problem



I made some triangles out of sticks. The sticks are all the same length.

I can make a triangle out of three sticks:

Image
Sticks and Triangles


What about four sticks? Five sticks? Six sticks?

I can make two different triangles out of ten sticks:

Image
Sticks and Triangles


Can you find any others?

What about triangles made from twenty sticks?

What about all the lengths in between?

Can you make any rules about the numbers of sticks that make the most triangles?