A333642 Number of regions in a polygon whose boundary consists of n+2 equally space 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

1,1

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.

Sequence in context: A072250 A220908 A057566 * A009303 A096586 A131128

Adjacent sequences: A333639 A333640 A333641 * A333643 A333644 A333645

Keyword

nonn

Author

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

Extensions

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

Status

approved