A335527 -id:A335527 - cp's OEIS frontend

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

Ethan Beihl, Nov 28 2016

nonn

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.

			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.

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)

3-Portolan numbers (equilateral triangle): A277402.
n-sected sides (not angles): A108914.
Cf. A277402, A335526 (vertices), A335527 (edges), A335528 (ngons).

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.

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

A335526 a(n) is the number of vertices formed by n-secting the angles of a square.

Original entry on oeis.org

4, 5, 32, 29, 96, 69, 192, 169, 324, 285, 492, 409, 696, 629, 936, 865, 1208, 1117, 1508, 1429, 1852, 1757, 2232, 2089, 2648, 2521, 3100, 2961, 3580, 3425, 4092, 3949, 4648, 4477, 5240, 5029, 5860, 5665, 6516, 6325, 7204, 7001, 7936, 7737, 8696, 8473, 9500

Offset: 1

Lars Blomberg, Jun 12 2020

nonn

See A278823 for illustrations.

Cf. A335411, A278823 (regions), A335527 (edges), A335528 (ngons).

A335528 Irregular table read by rows: T(n,k) = number of k-sided polygons formed by n-secting the angles of a square for k >= 3.

Original entry on oeis.org

0, 1, 4, 12, 16, 0, 0, 0, 1, 24, 4, 4, 48, 28, 8, 4, 4, 1, 52, 32, 80, 68, 28, 12, 0, 1, 96, 68, 20, 4, 128, 132, 36, 16, 8, 1, 156, 120, 24, 16, 192, 196, 60, 36, 0, 5, 200, 220, 28, 12, 244, 312, 80, 56, 0, 1, 296, 276, 60, 44, 360, 372, 112, 76, 12, 1

Offset: 1

Lars Blomberg, Jun 12 2020

See A278823 for illustrations.

			Table begins:
0, 1;
4;
12, 16, 0, 0, 0, 1;
24, 4, 4;
48, 28, 8, 4, 4, 1;
52, 32;
80, 68, 28, 12, 0, 1;
96, 68, 20, 4;
128, 132, 36, 16, 8, 1;
156, 120, 24, 16;
192, 196, 60, 36, 0, 5;
200, 220, 28, 12;
244, 312, 80, 56, 0, 1;
296, 276, 60, 44;
360, 372, 112, 76, 12, 1;

Cf. A335413, A278823 (regions), A335526 (vertices), A335527 (edges).

cp's OEIS Frontend

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

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A335526 a(n) is the number of vertices formed by n-secting the angles of a square.

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A335528 Irregular table read by rows: T(n,k) = number of k-sided polygons formed by n-secting the angles of a square for k >= 3.

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