A278823 - OEIS

%I #16 Jun 12 2020 06:17:27

%S 1,4,29,32,93,84,189,188,321,316,489,460,693,676,933,916,1205,1180,

%T 1505,1496,1849,1836,2229,2188,2645,2616,3097,3060,3577,3536,4089,

%U 4064,4645,4604,5237,5176,5857,5808,6513,6472,7201,7160,7933,7900,8693,8648,9497

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

%C 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:

%C 1. Solve for the coordinates of all intersections between lines within the square, recording multiplicity.

%C 2. Use an elementary Euler's-formula method as in Poonen and Rubinstein 1998 to turn the intersection-count into a region-count.

%H Lars Blomberg, <a href="/A278823/b278823.txt">Table of n, a(n) for n = 1..500</a>

%H Ethan Beihl, <a href="http://ethan.beihl.org/erb/creative-work/mathematical-visualization/">Pictures for some small n</a>

%H Lars Blomberg, <a href="/A278823/a278823.png">Coloured illustration for n=4</a>

%H Lars Blomberg, <a href="/A278823/a278823_1.png">Coloured illustration for n=5</a>

%H Lars Blomberg, <a href="/A278823/a278823_2.png">Coloured illustration for n=64</a>

%H Lars Blomberg, <a href="/A278823/a278823_3.png">Coloured illustration for n=65</a>

%H B. Poonen and M. Rubinstein (1998) <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:<a href="http://dx.doi.org/10.1137/S0895480195281246">10.1137/S0895480195281246</a>, arXiv:<a href="http://arXiv.org/abs/math.MG/9508209">math.MG/9508209</a> (typos corrected)

%F 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.

%e 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.

%Y 3-Portolan numbers (equilateral triangle): A277402.

%Y n-sected sides (not angles): A108914.

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

%K nonn

%O 1,2

%A _Ethan Beihl_, Nov 28 2016

%E a(24) and beyond from _Lars Blomberg_, Jun 12 2020