A333642 Number of regions in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and three equally spaced points along the diameter (a total of n+3 points). See Comments for precise definition.

2, 8, 20, 43, 80, 139, 224, 324, 510, 730, 992, 1373, 1820, 2187, 3040, 3844, 4720, 5916, 7220, 8498, 10472, 12463, 14570, 17278, 20150, 23130, 26964, 30961, 34688, 40265, 45632, 51138, 57970, 65008, 72322, 80979, 89984, 99197, 110240, 121570, 132896, 146818

Offset: 1

Author

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

Keywords

• nonn

Comments

A semicircular polygon with n+3 points is created by placing n+2 equally spaced vertices along the semicircle's arc (including the two end vertices). Also place three equally spaced vertices along the diameter; these are the same two end vertices plus one dividing the diameter. Now connect every pair of vertices by a straight line segment. The sequence gives the number of regions in the resulting figure.

Links

• Lars Blomberg, Table of n, a(n) for n = 1..100
• 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. A330914 (n-gons), A330911 (edges), A330913 (vertices), A333643, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

Extensions

a(21) and beyond from Lars Blomberg, May 03 2020