OFFSET
1,2
COMMENTS
m-Portolan numbers for m>3 (especially m even) are more difficult than m=3 (A277402) because Ceva's theorem doesn't immediately give us a condition for redundant intersections. The values for n <= 23 were found by brute force in Mathematica, as follows:
1. Solve for the coordinates of all intersections between lines within the square, recording multiplicity.
2. Use an elementary Euler's-formula method as in Poonen and Rubinstein 1998 to turn the intersection-count into a region-count.
LINKS
Lars Blomberg, Table of n, a(n) for n = 1..500
Ethan Beihl, Pictures for some small n
Lars Blomberg, Coloured illustration for n=4
Lars Blomberg, Coloured illustration for n=5
Lars Blomberg, Coloured illustration for n=64
Lars Blomberg, Coloured illustration for n=65
B. Poonen and M. Rubinstein (1998) The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:10.1137/S0895480195281246, arXiv:math.MG/9508209 (typos corrected)
FORMULA
For n = 2k - 1, a(n) is close to 18k^2 - 26k + 9. For n = 2k, a(n) is close to 18k^2 - 26k + 12. The residuals are related to the structure of redundant intersections in the figure.
EXAMPLE
For n=3, the 4*(3-1) = 8 lines intersect to make 12 triangles, 8 kites, 8 irregular quadrilaterals, and an octagon in the middle. The total number of regions a(3) is therefore 12+8+8+1 = 29.
CROSSREFS
KEYWORD
nonn
AUTHOR
Ethan Beihl, Nov 28 2016
EXTENSIONS
a(24) and beyond from Lars Blomberg, Jun 12 2020
STATUS
approved