The first few Lucas numbers are
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349
Attached code is a way to visualize Lucas numbers geometrically for n>2
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Graph ( n )-number of nodes on a unit circle.
- Place ( n ) nodes equidistantly around a unit circle.
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Count the edges around the unit circle.
- This will always be ( n ), corresponding to the number of nodes.
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Count the overall shape itself (or consider the empty set).
- This contributes 1 to the count, representing the entire ( n )-gon or the empty set.
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Remove the counted edges and overall shape, then count the number of unique shapes that can be made with the remaining edges (essentially, diagonals).
- These are the edges that are not part of the original ( n )-gon. They form unique shapes within the ( n )-gon.
By following these steps, you end up with a count that aligns with the Lucas numbers ( L_n ) for the respective ( n )-gons.
