A333519 -id:A333519 - cp's OEIS frontend

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

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

nonn

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.

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.

Cf. A330914 (n-gons), A330911 (edges), A330913 (vertices), A333643, A333519, A007678, A290865, A092867, A331452, A331929, A331931.

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

A334458 Number of vertices in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

Original entry on oeis.org

1, 4, 12, 39, 125, 271, 609, 1076, 1884, 2950, 4642, 6541, 9607, 12969, 17505, 23034, 30294, 37888, 48488, 59404, 73506, 88779, 108077, 127412, 153000, 178514, 210366, 242961, 283243, 322120, 373147, 422454, 482442, 542604, 615300, 685885, 773189, 857791

Offset: 0

Lars Blomberg, May 01 2020

nonn

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

Lars Blomberg, Table of n, a(n) for n = 0..50

Cf. A333519, A334459

A334459 Number of edges in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

Original entry on oeis.org

0, 5, 24, 86, 265, 582, 1260, 2235, 3861, 6055, 9416, 13366, 19409, 26296, 35408, 46569, 60945, 76527, 97432, 119703, 147753, 178670, 216844, 256401, 306825, 358719, 421848, 487960, 567617, 647259, 747596, 847765, 966609, 1088541, 1232280, 1376016, 1548265

Offset: 0

Lars Blomberg, May 01 2020

nonn

Lars Blomberg, Table of n, a(n) for n = 0..50

Cf. A333519, A334458

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.

Original entry on oeis.org

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

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

nonn

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.

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.

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

More terms from Lars Blomberg, Apr 20 2020

cp's OEIS Frontend

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.

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A334458 Number of vertices in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

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A334459 Number of edges in a polygon whose boundary consists of n+2 equally spaced points around a semicircle and n+2 equally spaced points along the diameter (a total of 2n+2 points). See Comments for precise definition.

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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.

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