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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A semi-circular 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 A165751

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

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Last modified September 25 23:23 EDT 2020. Contains 337346 sequences. (Running on oeis4.)