A333643 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around the arc of a semicircle. See Comments for precise definition.

1, 4, 11, 25, 50, 91, 154, 234, 375, 550, 769, 1079, 1456, 1783, 2500, 3196, 3987, 5016, 6175, 7348, 9086, 10879, 12836, 15250, 17875, 20682, 24129, 27811, 31419, 36425, 41416, 46664, 52921, 59500, 66489, 74481, 82954, 91807, 102050, 112750, 123700, 136654

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 31 2020

Keywords

• nonn

Comments

A semicircular polygon with n+2 points is created by placing n+2 equally spaced vertices along a semicircle's arc, which includes the two end vertices. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Note that there is a curious relationship between the terms of this sequence and the number of regions in the 'general position' polygon given in A006522. They are a match except for every third term starting at a(8) = 234. Examining the images for n = 8,11,14,17 shows that these polygons have interior points at which three or more lines intersect, while the other n values have no such intersection points. Such multi-line intersection points will reduce the number of regions as compared to the general position polygon which has no multi-line intersection points. This is reflected by the terms in this sequence being lower than the corresponding value in A006522 for n = 8,11,14,... . Why every third value of n in this sequence starting at n = 8 leads to polygons having multiple line intersection points while other values of n do not is currently not known.

Links

• Lars Blomberg, Table of n, a(n) for n = 1..80
• Scott R. Shannon, Illustration for n = 2.
• Scott R. Shannon, Illustration for n = 3.
• Scott R. Shannon, Illustration for n = 4.
• Scott R. Shannon, Illustration for n = 5.
• Scott R. Shannon, Illustration for n = 7.
• Scott R. Shannon, Illustration for n = 10.
• Scott R. Shannon, Illustration for n = 12.
• Scott R. Shannon, Illustration for n = 15.
• Scott R. Shannon, Illustration for n = 17.
• Scott R. Shannon, Illustration for n = 19.
• Scott R. Shannon, Illustration for n = 20.
• Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
• Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.
• Scott R. Shannon, Illustration for n = 19 with random distance-based coloring.
• Scott R. Shannon, Illustration for n = 20 with random distance-based coloring.
• Wikipedia, Semicircle.

Crossrefs

Cf. A333642, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

Extensions

More terms from Lars Blomberg, Apr 20 2020