The Order of Lines

Arrange N points evenly around a circle. Label them A,B,C... clockwise. Starting at point A, draw a continuous path of lines, visiting each point once and finishing at point A. For N=3, there is only one figure which can be drawn (a triangle), regardless of the order the points are visited. For N=4, there are two different figures: a square (ABCDA), and a cross with two sides (ABDCA). [Although a rotated version of the second figure can be drawn (ACBDA), we consider them identical.]

Disregarding rotations and reflections, how many different figures can be made for

  1. N=5?
  2. N=6?
  3. N=7?

Source: Based on a problem from The Little Giant Encyclopedia of Puzzles.

Solutions were received from Sudipta Das, Hareendra Yalamanchili, Al Zimmermann, and Becky Albert. Al's results follow: 
 N     Number of Figures
 ------------------------
 5               4
 6              12
 7              39
 8             202
 9           1,219
10           9,468

5 lines:



6 lines:



7 lines:


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