A278823 4-Portolan numbers: number of regions formed by n-secting the angles of a square.

1, 4, 29, 32, 93, 84, 189, 188, 321, 316, 489, 460, 693, 676, 933, 916, 1205, 1180, 1505, 1496, 1849, 1836, 2229, 2188, 2645, 2616, 3097, 3060, 3577, 3536, 4089, 4064, 4645, 4604, 5237, 5176, 5857, 5808, 6513, 6472, 7201, 7160, 7933, 7900, 8693, 8648, 9497

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

3-Portolan numbers (equilateral triangle): A277402.

n-sected sides (not angles): A108914.

Cf. A277402, A335526 (vertices), A335527 (edges), A335528 (ngons).

Sequence in context: A327436 A339266 A247278 * A042597 A255635 A280854

Adjacent sequences: A278820 A278821 A278822 * A278824 A278825 A278826

Keyword

nonn

Author

Ethan Beihl, Nov 28 2016

Extensions

a(24) and beyond from Lars Blomberg, Jun 12 2020

Status

approved